This case is, in spirit, a companion to our recent decision in Imani v. Pollard, No. 14-3407, 826 F.3d 939, 2016 WL 3434673 (7th Cir. June 22, 2016). It, too, raises the question whether a criminal defendant’s right to self-representation — acknowledged by the Supreme Court in Faretta v. California, 422 U.S. 806, 95 S.Ct. 2525, 45 L.Ed.2d 562 (1975) — was infringed. In our case, Joseph Jordan was on trial for reckless homicide in Wisconsin. He moved to waive counsel and represent himself because he feared that his court-appointed attorney was not up to the job. The court denied his motion. What happened at trial, in Jordan’s view, vindicated his fears: his attorney failed to object to a series of improper statements during the state’s closing argument when the prosecutor vouched for the credibility of a witness. Jordan now seeks habeas corpus relief, either on the basis of the denial of his Faretta right or his failure to receive the assistance of counsel to which the Sixth Amendment entitles him. We conclude that he is entitled to proceed on the latter ground, and thus we reverse and remand for a hearing under 28 U.S.C. § 2254(e)(2).
*841I
In 2003, a Wisconsin state trial court convicted Jordan of one count of first-degree reckless homicide, three counts of first-degree endangerment, and one count of being a felon in possession of á firearm. The charges stemmed from the shooting death of David Robinson. Robinson was sitting in a car with three other people when he was killed by shots fired from a passing car. The state contended that Jordan was the shooter.
The prosecution’s theory was that as Jordan sat in the passenger seat of the passing car, he reached across the driver — Michael Blake Jones (“Blake”) — and fired at Robinson. Eyewitnesses presented conflicting accounts. One passenger in Robinson’s car identified Blake as the shooter. The driver of Robinson’s car was unable to identify either Blake or Robinson as the shooter, but he admitted that he might initially have told the police that Blake was the shooter. Another passenger in Robinson’s car, Tashanda Washington, identified Jordan as the shooter. A defense witness testified that Washington had previously admitted to her that Blake was the shooter, but that Washington and Blake had agreed to pin the crime on Jordan.
In some ways, this jumbled eyewitness testimony was just a sideshow. That was so because Jordan (supposedly) signed a confession that he was the shooter. The confession took center stage at the trial, where the parties hotly contested how it came to be. As Jordan told the story, he was interrogated for 13 hours over a 27-hour period, during which he steadfastly maintained his innocence. At the end, he says, his interrogators presented him with a document and falsely told him it “only sa[id] what we talked about” and that he could go home if he signed it. In fact, it was a written confession, which Jordan signed without reading — because, as those two detectives knew and as both parties now agree, Jordan is nearly illiterate. The government’s witnesses had a different recollection. They said that Jordan confessed during the interrogation, that the detectives wrote up an accurate statement of his oral confession, and that Jordan then signed it because, as the document states, “he wanted to tell the truth about his involvement in this incident.” Everyone expected that the trial would turn on which story the jury believed.
Jordan’s dissatisfaction with his attorney, Russell Bohach, long predated the trial. He repeatedly complained about Bo-hach, telling the court that Bohach was not meeting with him or investigating leads properly. After the court postponed the initial trial date, Jordan reiterated his concerns at an evidentiary hearing held the day before the new trial date. While the hearing was underway, Jordan asked the court to do one of three things: appoint new counsel, delay the trial to allow Bo-hach to conduct further research, or allow Jordan to represent himself. The court immediately rejected the first two options, but it engaged in an extended colloquy with Jordan about the third.
The court first canvassed Jordan’s background and experience. Jordan stated that he had an eighth-grade education, but only a fourth-grade reading ability. He had experience with being charged, but not with a trial. He confirmed that he understood the elements of the charges against him and correctly recited the potential sentence if convicted. He also said that he understood his constitutional right to counsel and the role of counsel — including the fact that if he were to represent himself, his stand-by counsel could not make his case for him. He acknowledged that without counsel, he would not know to make certain legal arguments, including objections to evidence or jury instructions. The court then ruled that he was competent to *842waive counsel and allowed him to do so. It commented that Jordan “certainly appears to me to be of average capability generally” and that he “seems alert and reasonably bright and [to] have some general understanding of what’s going on here.” While the court expressed concern about Jordan’s limited literacy, it stated “that at least under the circumstances of this case, which is ultimately a factual case and not a document’s [sic] case, that that limitation should not prevent him from representing himself.”
We would not be here if matters had rested there. But they did not: later that day, the court reconsidered. It returned to the interrupted evidentiary hearing, this time with Jordan representing himself. During the hearing,- the court reviewed a police report. This prompted it to worry that Jordan’s limited literacy would prevent him from using documents provided in discovery, including police reports, “in any meaningful way at trial.” After discussing the documents and taking a recess to allow Jordan to read some documents on his own, the court asked Jordan to read ' a document aloud and explain whether he understood it. Jordan responded that he only “somewhat” understood it, but that he was not concerned by this because his theory of defense did not depend on any documents. In fact, he told the court, he had already written down the questions that he wanted to ask of various witnesses. The court was not reassured and decided that it had to reverse its earlier ruling. It explained that given Jordan’s limited literacy and education, he would “not [be] able to effectively represent [himself] and present a meaningful defense in this case.” While Jordan was “competent to proceed to trial,” he was “not competent to represent [himself] in a case of this nature” and so it ordered Bohach to represent Jordan at trial.
At trial, Bohach gave Jordan reason to be displeased. The key issue, recall, was whose account of the supposed confession the jury would believe. In its closing argument, the prosecution made a series of statements vouching for the detectives’ credibility and urged the jury to bear in mind who had the most to lose — Jordan or the prosecutor. The jury got the message and convicted Jordan on all counts.
Jordan represented himself in a post-trial motion to the trial court, where he raised the two arguments now before us. The trial court denied his motion in a written order dated September 17, 2004. He continued to represent himself on direct appeal, where he raised the same points (among others). He was unsuccessful in the Wisconsin Court of Appeals, which adopted the trial court’s post-trial order, and the Wisconsin Supreme Court denied review. Jordan then wended his way through the state post-conviction review process, where he raised other points, again to no avail.
Jordan then filed a petition seeking federal habeas corpus relief pursuant to 28 U.S.C. § 2254. He alleged that his Sixth Amendment rights were violated by the Wisconsin court’s refusal to allow him to represent himself at trial, and later by Bohach’s ineffective assistance of counsel. The district court denied his petition and denied a certificate of appealability. We granted a certificate of appealability on both contentions.
II
A
Jordan first argues that Wisconsin denied him his right under the Sixth and Fourteenth Amendments to represent himself. See Faretta, 422 U.S. 806, 95 S.Ct. 2525. We review the district court’s denial of his habeas corpus petition de novo. *843Campbell v. Smith, 770 F.3d 540, 546 (7th Cir. 2014). Under the Antiterrorism and Effective Death Penalty Act of 1996 (AEDPA), however, the applicable legal standard is one that calls for great deference to the state court. Where the state court has made a decision on the merits, we may grant relief only if that decision was “contrary to, or involved an unreasonable application of clearly established Federal law” as determined by the Supreme Court. 28 U.S.C. § 2254(d)(1).
We apply this standard to the decision of the last state court to rule on the merits of the petitioner’s claim. Ylst v. Nunnemaker, 501 U.S. 797, 111 S.Ct. 2590, 115 L.Ed.2d 706 (1991). This may be a summary decision. See Harrington v. Richter, 562 U.S. 86, 99, 131 S.Ct. 770, 178 L.Ed.2d 624 (2011). Here, the last decision on the merits was the one handed down by the Wisconsin Court of Appeals in August 2005. Because that court explicitly adopted the trial court’s September 2004 order, we also consider the trial court’s order as part of the “last reasoned opinion” of the Wisconsin courts. See Stitts v. Wilson, 713. F.3d 887, 891 (7th Cir. 2013).
Although the Wisconsin court came close to making an unreasonable application of the Faretta line of cases, these decisions depend heavily on the facts before the court, and on this record we cannot say that the state court crossed the line. The question on which this case turns relates to the distinction between competence to make decisions, and ability to handle a trial. The Faretta rule has been around since 1975. See 422 U.S. at 834-36, 95 S.Ct. 2525. In recognizing the right of self-representation, the Supreme Court said there that while a defendant’s waiver of the right to counsel must be made “knowingly and intelligently,” that defendant’s “technical legal knowledge” is “not relevant.” Id. at 835-36, 95 S.Ct. 2525.
Wisconsin established its standard for waiving counsel in Pickens v. State, 96 Wis.2d 549, 292 N.W.2d 601 (1980). In State v. Klessig, 211 Wis.2d 194, 564 N.W.2d 716 (1997), it confirmed that intervening U.S. Supreme Court precedent had not required any change in Wisconsin law. The Wisconsin Supreme Court took the position in Klessig, in some tension with Faretta, that “competency to stand trial is not the same as competency to proceed pro se.” Klessig, 564 N.W.2d at 726 (internal quotation marks omitted). When determining whether to permit a defendant to represent herself, the trial court must “consider factors such as ‘the defendant’s education, literacy, fluency in English, and any physical or psychological disability which may significantly affect his ability to communicate a possible defense to the jury.’ ” Id. at 724 (quoting Pickens, 292 N.W.2d at 611).
The state trial court faithfully applied the Klessig standard. Although it found Jordan was “alert and reasonably bright” and of “average capability,” it determined that he could not proceed pro se because “he lacked the capacity to present a meaningful defense” or “effectively] repre-sente ] himself.” The first question is whether, as Jordan contends, this inquiry is actually “contrary to” Faretta, as 28 U.S.C. § 2254(d)(1) uses the term. Put differently, does Faretta flatly forbid an inquiry into the defendant’s ability to represent himself once the court has determined that the defendant is competent to waive his right to counsel?
We would be inclined to agree with Jordan if Faretta were the Supreme Court’s last word on the subject, but it is not. Despite the statement in Faretta that a defendant’s technical legal ability to represent herself is not relevant to the decision whether to permit waiver of counsel, the Supreme Court has not banned all inquiry *844into this subject. It has delineated some ways -in which states may limit the right to self-representation. For example, in Faret-ta itself the Court recognized that a trial court may forbid a defendant from representing herself if she is disruptive in the courtroom or engages in misconduct. 422 U.S. at 835 n.46, 95 S.Ct. 2525. Later it held that a state court may forbid a defendant from representing herself on direct appeal. Martinez v. Ct. of Appeal of Cal., Fourth Appellate Dist., 528 U.S. 152, 163, 120 S.Ct. 684, 145 L.Ed.2d 597 (2000).
In Godinez v. United States, the Supreme Court elaborated on what states may require of a defendant who wishes to represent herself. 509 U.S. 389, 113 S.Ct. 2680, 125 L.Ed.2d 321 (1993). There the Court held that when a defendant wishes to enter a guilty plea pro se, due process does not require that she meet any higher level of competency beyond the competency to stand trial and to waive her right to a trial knowingly and intelligently. Id. at 402, 113 S.Ct. 2680. But the Court also noted, perhaps trimming Faretta a bit, that “States are free to adopt competency standards that are more elaborate” than the minimum standard the Constitution requires. Id. It did so, however, as part of a holding rejecting the proposition that “the competency standard for pleading guilty or waiving the right to counsel is higher than the competency standard for standing trial.” Id. at 391, 113 S.Ct. 2680. It held that there is only one competency standard. Thus, its comment that states might wish to adopt more elaborate standards must be understood as a comment about evidence, not a comment about the governing legal question. Whatever the standard is, Godi-nez held that it applies across the board.
The Supreme Court then qualified Godinez in Indiana v. Edwards, in which it upheld Indiana’s requirement that a defendant who wishes to represent himself must have the “mental capacity” to “conduct his trial,” which is a higher standard than competency to stand trial. 554 U.S. 164, 174, 128 S.Ct. 2379, 171 L.Ed.2d 345 (2008). In Edwards, the defendant (whose motion to represent himself was denied) “suffered from schizophrenia,” which caused “delusions and ... marked difficulties in thinking.”.Id. at 168-69, 128 S.Ct. 2379. The Court held that the Constitution permits a state to limit the self-representation right for a defendant who “lacks the mental capacity to conduct his trial defense unless represented.” Id. at 174, 128 S.Ct. 2379. This holding seems to have broken the Godinez rule eschewing multiple competence standards for multiple purposes. Although Edwards was decided two years after Jordan’s criminal case became final on direct appeal in 2006, we may consider it here because Warden Hepp has raised it. See Moore v. Anderson, 222 F.3d 280, 285 (7th Cir. 2000) (nonretroactivity only “favors the state’s interest in finality” and thus the state may waive it).
In Imani, we characterized Edwards as identifying “a narrow class of cases in which a defendant may not be competent to represent himself[.]” 826 F.3d at 946-47, 2016 WL 3434673 at *5. There was no evidence of such circumstances in Imani, where the trial judge paid little heed to Imani’s level of education and placed heavy weight on the rationality of Imani’s decision. Id. 826 F.3d at 941-43, 2016 WL 3434673 at *1-2. It is permissible, for instance, to deny self-representation to a defendant who “sufferfs] from mental illness or mental impairment.” Id. 826 F.3d at 946, 2016 WL 3434673 at *5. Imani also recognized, however, that there would be “gray-area” defendants. States may not resolve. doubts against self-representation and remain faithful to Faretta. In Imani, the defendant’s “abilities were close enough to Faretta’s as to be indistinguishable.” Id. In that situation, we found that the Wisconsin courts unreasonably applied *845Faretta when they denied Imani his right to represent himself.
If it were up to us, we would find that Jordan’s Faretta rights were violated. Unfortunately, a great many adults in the United States are illiterate. The National Center for Education Statistics, which is in the Office of Educational Research and Improvement of the U.S. Department of Education, reported in 2002 that 21 to 23% of adults “demonstrated skills in the lowest level of prose, document, and quantitative' proficiencies.” • See U-S. Dep’t of Eduo., Adult Literacy in America, Executive Summary at xvi (2002), https://nces.ed.gov/ pubs93/93275.pdf. The numbers do not seem to be much better now. See The U.S. Illiteracy Rate Hasn’t Changed in 10 Years, Huffington Post (Sept. 6, 2013), http://www.huffingtonpost.com/2013/09/06/ illiteracy-rate_n_3880355.html [https:// perma.cc/LK5S-JDA6] (estimating that 14% of the population, or 32 million adults, cannot read at all, and 21% read below a fifth-grade level) (referencing U.S. Dep’t of Educ., 2012/2014 First Look National Supplement, 5-6 (2016), http://nces.ed.gov/ pubs2016/2016039.pdf). It is hard to imagine a person with limited literacy navigating his or her way through the court system without the assistance of a lawyer, even though it is likely that the person would be able to help his or her lawyer put on their defense. We see no hint that the Supreme Court was talking about this vast population in Edwards; instead, it was addressing the more serious problem of the mentally ill or mentally impaired person, who cannot handle matters himself and who needs a lawyer almost in the capacity of a guardian.
In our ease, Jordan demonstrated a good understanding of his case, and he had devised a defense strategy that he wanted to use — one that did not depend on written documents. That may have been foolish, but it was no more foolish than the approach taken by many who choose the self-representation route. With the same concern courts always feel when an unsophisticated defendant insists on his Faretta rights, we would say that he had such a right, and it was denied when the trial court re-visited its initial ruling.
But AEDPA does not permit us to apply our independent assessment of this case. Instead, we must decide whether the state court’s decision was so far beyond the pale that it was “unreasonable.” See Richter, 562 U.S. at 103, 131 S.Ct. 770. The fact that the Supreme Court itself recognized that certain forms of mental disability deserved special consideration suggests that the approach of the Wisconsin courts was not so utterly without support that it cannot stand. Jordan’s problem went well beyond the lack of knowledge of court procedure or an ability to make strategic judgments. Despite his bravado, Jordan was unlikely to be able to avoid confronting the written evidence in his case — evidence that was functionally unavailable to him because of his near-illiteracy. The state court thought that Jordan could not put on a defense without another person’s assistance, and that other person had to be a member of the bar. We feel compelled by AEDPA to hold that this was not an unreasonable interpretation of Faretta and Godinez, and thus neither contrary to those decisions nor an unreasonable application of their holdings.
B
Jordan may also prevail on his Faretta theory if the Wisconsin court’s decision “was based on an unreasonable determination of the facts in light of the evidence.” 28 U.S.C. § 2254(d)(2). He has attempted to meet that standard, but all he has shown is that another court might have *846viewed the facts differently, and that is not enough.
The trial court reached its factual conclusions based on Jordan’s own statements and the court’s observations of Jordan’s reading ability. Jordan admitted that he read at a fourth-grade level. He read the police report aloud to the court with dismal results, and acknowledged that he only “somewhat” understood it. He did not deny that he would be unable to use most of the documents produced for trial; he said instead that the practical absence of the documents would not hinder his ability to present a defense. It was reasonable for the court to infer that Jordan would not be able to use the documents, and that this would prevent him from effectively representing himself. This determination did not “lie[ ] against the clear weight of the evidence.” See Ward v. Sternes, 334 F.3d 696, 704 (7th Cir. 2003) (explaining the § 2254(d)(2) standard) (citing Hall v. Washington, 106 F.3d 742, 749 (7th Cir. 1997)).
Ill
We now turn to Jordan’s ineffective-assistance-of-counsel claim. Once the court imposed Bohach on Jordan, Jordan had the right to a performance that was sufficiently competent to “funetion[ ] as the ‘counsel’ guaranteed ... by the Sixth Amendment.” See Strickland v. Washington, 466 U.S. 668, 687, 104 S.Ct. 2052, 80 L.Ed.2d 674 (1984). Counsel is constitutionally deficient if her assistance falls below “an objective standard of reasonableness” and these errors “prejudice!] the defendant.” Id. at 687-88, 104 S.Ct. 2052. This is not an easy standard to satisfy. “Counsel is strongly presumed to have rendered adequate assistance and made all significant decisions in the exercise of reasonable professional judgment.” Id. at 690, 104 S.Ct. 2052. Sufficient “prejudice” requires that “there [be] a reasonable probability that, but for counsel’s unprofessional errors, the result of the proceeding would have been different.” Id. at 694, 104 S.Ct. 2052. Section 2254(d)(1)’s deferential standard of review of the state court’s decision, combined with Strickland’s presumption that counsel acted reasonably, leads our review to be “doubly” deferential toward the state court’s decision on this claim. Richter, 562 U.S. at 105, 131 S.Ct. 770.
A
The Supreme Court has long held that due process, applicable to the states through the Fourteenth Amendment, requires fair legal procedure, and that “[a]mong these ‘legal procedures’ is the requirement that the jury’s verdict be based on evidence received in open court, not from outside sources.” Sheppard v. Maxwell, 384 U.S. 333, 351, 86 S.Ct. 1507, 16 L.Ed.2d 600 (1966); see also Marshall v. United States, 360 U.S. 310, 313, 79 S.Ct. 1171, 3 L.Ed.2d 1250 (1959); Skilling v. United States, 561 U.S. 358, 378-81, 130 S.Ct. 2896, 177 L.Ed.2d 619 (2010) (out-of-court evidence in the form of press coverage can interfere with a defendant’s due process right to a fair trial). In Parker v. Gladden, 385 U.S. 363, 87 S.Ct. 468, 17 L.Ed.2d 420 (1966) (per curiam), the Court granted habeas corpus relief to an Oregon prisoner because the bailiff who was escorting the jury expressed his own opinions about the defendants’ guilt in separate statements to at least two jurors.
One aspect of the need to confine the jury to the evidence that was properly admitted to trial is the rule providing that a prosecutor may not urge a jury to base its decision on information known to the prosecutor but not presented at trial. See Berger v. United States, 295 U.S. 78, 88, 55 S.Ct. 629, 79 L.Ed. 1314 (1935). Although Berger itself involved a federal prosecution, there is no meaningful distinction between the Fifth Amendment’s *847due process clause and that of the Fourteenth for this purpose. The Court was concerned that the prosecutor there had invited the jury to rely on evidence within the personal knowledge of the prosecutor-evidence that went beyong the record. After reproducing excerpts from the improper argument, the Court said “[i]n these circumstances prejudice to the cause of the accused is so highly probable that we are not justified in assuming its nonexistence.” Id. at 89, 55 S.Ct. 629. This is the language of due process, not mere supervision.
The constitutional basis of this concern is not diminished by the fact that vouching, like almost all trial error, is subject to the plain error rule. See United States v. Young, 470 U.S. 1, 18, 105 S.Ct. 1038, 84 L.Ed.2d 1 (1985). The Court expresses its concerns about vouching in constitutional terms: “Due process requires that the accused receive a trial by an impartial jury free from outside influences.” Nebraska Press Ass’n v. Stuart, 427 U.S. 539, 553, 96 S.Ct. 2791, 49 L.Ed.2d 683 (1976). The jury may not decide the case based on evidence that never made it into the record, either real or imagined. The right to a trial by jury includes the right to the jury’s own decision, not a decision dictated or unduly influenced by the prosecutor. Young, 470 U.S. at 18-19, 105 S.Ct. 1038 (citing Berger, 295 U.S. at 88-89, 55 S.Ct. 629).
In Jordan’s case, the prosecutor had this to say during his closing argument:
Now, the big question here is the credibility. Who do you believe? ... Somebody’s lying. Who is it? [The detective’s] going to put her whole career and her future on the line for this case? She does this everyday. She’s investigating homicide cases everyday for years. Who has the most to lose based on your verdict in this case? Her or him? ... It boils down to credibility.
The prosecutor then repeated, “[w]ho has the most to lose here? Her or him? Keep that in mind when you evaluate his testimony.”
This is a textbook case of improper vouching. The prosecutor engaged in one of the forms of argumentation that the Supreme Court repeatedly has identified as improper: implying that the jury should believe a witness based on evidence that was not presented to the jury. See Berger, 295 U.S. at 88, 55 S.Ct. 629; Young, 470 U.S. at 18, 105 S.Ct. 1038. By arguing that the detective would lose her job by giving false testimony, the prosecutor “eonvey[ed] the impression that evidence not known to the jury"-namely, that the detective would face career repercussions for false testimony-"support the charges against the defendant.” See Young, 470 U.S. at 18, 105 S.Ct. 1038. The prosecutor left no room for doubt about his message: he stated three times that she has more to lose than the defendant, and is therefore more likely to be telling the truth.
As Young recognizes, this is the type of misconduct that the Supreme Court forbade in Berger. There, a prosecutor stated his personal opinion that a witness was lying about whether she knew the defendant. The prosecutor implied that he knew the witness was lying; that evidentiary rules did not permit the jury to hear that evidence; and that the jury should instead just trust him. Berger, 295 U.S. at 86-88, 55 S.Ct. 629. The Supreme Court pinpointed the troubling aspect of this argument: “The jury was thus invited to conclude that the witness ... knew [the defendant] well but pretended otherwise; and that this was within the personal knowledge of the prosecuting attorney.” Id. at 88, 55 S.Ct. 629. The same concern arises in Jordan’s case. The prosecutor relied on evidence not *848in the record but that appeared to be within his personal knowledge: that the detective would lose her job if she wasn’t telling the truth.
Due process therefore forbids a prosecutor to urge a jury to rely on evidence that is not in the record, whether that evidence is from newspaper accounts, the Internet, or the prosecutor’s own mouth. It requires the jury to be left alone to do its own job, evaluating the evidence the trial judge admitted, and coming to its own independent conclusion (as opposed to one dictated by the prosecutor).
. In Jordan’s trial, Bohach failed to object to any of the prosecutor’s improper statements. Our first question is whether that failure rendered his performance ineffective under Strickland. The Supreme Court held in Wiggins v. Smith, 539 U.S. 510, 123 S.Ct. 2527, 156 L.Ed.2d 471 (2003), that “the deference owed to ... strategic judgments” depends on “the adequacy of the investigations supporting those judgments.” Id. at 521, 123 S.Ct. 2527. If counsel made a choice after “thorough investigation of law and facts relevant to plausible options,” that choice.is “virtually unchallengeable,” id. (citation omitted), but a decision made for no strategic reason at all does not command any deference.
The Warden suggests one possible legitimate explanation for Bohach’s failure to object to the improper vouching: perhaps, he argues, Bohach remained silent in order to avoid drawing the jury’s attention to the prosecutor’s statements. If that were so, then Bohach’s performance might not have been objectively unreasonable conduct. The problem is that there is no evidence in the record to support the Warden’s speculation. Apparently no one asked Bohach why he did not object. And the Warden’s counsel conceded at oral argument that the state court never held a hearing or made a factual finding on this issue. (In Wisconsin this is known as a Machner hearing, after State v. Machner, 92 Wis.2d 797, 285 N.W.2d 905 (Wis. Ct. App. 1979). The post-conviction court held a Machner hearing on a different issue in 2009, but did not touch this one). Indeed, the warden’s counsel admitted that were this issue before a state court, a Machner hearing would be necessary.
B
Despite these indicia of deficient performance, we cannot say on this record that Jordan has (or has not) satisfied this part of the Strickland inquiry. Moreover, even if he can show deficient performance, he must also demonstrate prejudice. Before returning to the performance question, we therefore turn to the prejudice issue. (If Jordan cannot show prejudice, there would be no need to resolve the performance question, as both must be shown in order to prevail under Strickland. 466 U.S. at 687, 104 S.Ct. 2052.)
A defendant is prejudiced when there is “a reasonable probability that, but for counsel’s unprofessional errors, the result of the proceeding would have been different.” Strickland, 466 U.S. at 694, 104 S.Ct. 2052. Because the state court addressed the question of prejudice (albeit briefly), we apply AEDPA’s deferential standard to this question.
The Supreme Court has stated that when a prosecutor improperly vouches for a witness’s credibility, and the case is not otherwise a strong one, “prejudice to the cause of the accused is so highly probable that we are not justified in assuming its nonexistence.” Berger, 295 U.S. at 89, 55 S.Ct. 629. This is such a ease. On this record, the prosecutor’s improper vouching for the credibility of one of the detectives went to the heart of the matter. Had defense counsel taken steps to cure that error, there is at least a “reasonable proba*849bility"-that is, "a probability sufficient to undermine confidence in the outcome,” see Richter, 562 U.S. at 104, 131 S.Ct. 770-that the result of the proceeding would have been different.
We recognize that the trial court instructed the jury that “the words of the attorneys are not the evidence in this case and their arguments and conclusions that they’re entitled to express at this stage are not evidence and must not be considered by you as evidence.” But this instruction did not identify the prosecutor’s remarks as improper statements that should be disregarded, for the obvious reason that those remarks had not yet been made. Nor was this or any other instruction given contemporaneously with, or immediately after, the prosecutor’s inappropriate comments. We cannot assume that a prompt objection, followed by a curative instruction, would have been ineffective; indeed, a prompt objection would have cut off a good part of the vouching. When the whole case turns on witness credibility, standing silent while the state vouches for its witnesses cannot be justified by reliance on a generic, non-contemporaneous instruction. See Donnelly v. DeChristoforo, 416 U.S. 637, 644, 94 S.Ct. 1868, 40 L.Ed.2d 431 (1974) (“some occurrences at trial may be too clearly prejudicial for such a curative instruction to mitigate their effect”); see also Goodman v. Bertrand, 467 F.3d 1022, 1030-31 (7th Cir. 2006); Earls v. McCaughtry, 379 F.3d 489, 495-96 (7th Cir. 2004); Cossel v. Miller, 229 F.3d 649, 655-56 (7th Cir. 2000); Hodge v. Hurley, 426 F.3d 368, 385 (6th Cir. 2005). The state trial court’s finding (adopted by the Court of Appeals) that counsel’s failure to object was not prejudicial is an unreasonable finding in the context of this case.
C
Because we have found prejudice, we must return to the performance question. As we have indicated, this record does not reveal whether Bohach.had any strategic reason for his silence. The answer to this.question may dictate the result for Jordan’s petition, and so we must remand for an evidentiary hearing under 28 U.S.C. § 2254(e)(2)(A)(ii). Section 2254(e)(2)(A)(ii) permits the district court to conduct an evidentiary hearing in limited circumstances: namely, when the state court record does not contain sufficient factual information to adjudicate a claim, and “the factual predicate could not have been previously discovered through the exercise of due diligence.” Here, the state court record does not contain this inform,ation, nor could the factual predicate have been discovered previously because the state court never held a Machner hearing. Jordan therefore satisfies the requirements of section 2254(e)(2)(A)(ii).
This procedure does not run afoul of Cullen v. Pinholster, 563 U.S. 170, 131 S.Ct. 1388, 179 L.Ed.2d 557 (2011). In Cullen, the Supreme Court held that “review under § 2254(d)(1) is limited to the record that was before the state court that adjudicated the claim on the merits.” Id. at 181, 131 S.Ct. 1388. We already have gone through the (d)(1) process, which led us to conclude that in the absence of evidence to support Warden Hepp’s counterargument, the Wisconsin Court of Appeals unreasonably applied Strickland when it held that Bohach’s performance did not fall below an objective standard of performance. That alone, however, does not entitle Jordan to relief. The Warden gets one more chance, under (e)(2), to demonstrate that Bohach had a strategic reason for his failure to object to the vouching.
Holding a hearing under section 2254(e)(2) in this situation is what Cullen envisioned. As Justice Breyer explained in his separate opinion, “if the state-court rejection assumed the habeas petitioner’s facts (deciding that, even if those facts *850were true, federal law was not violated), then (after finding the state court wrong on a [§ 2254(d) ] ground), a[ ] [§ 2254(e) ] hearing might be needed to determine whether the facts alleged were indeed true.” Cullen, 563 U.S. at 205, 131 S.Ct. 1388 (Breyer, J., concurring in part in dissenting in part); see also Campbell v. Reardon, 780 F.3d 752, 772 (7th Cir. 2015) (explaining when a § 2254(e) hearing is appropriate after Cullen); Taylor v. Grounds, 721 F.3d 809, 824-25 (7th Cir. 2013) (same). As Warden Hepp’s counsel acknowledged at oral argument, the state court never made any factual finding on this question to which we could defer. Because the petitioner has alleged facts that would make Bohach’s conduct objectively unreasonable under Strickland and the state’s contrary ruling unreasonable under section 2254(d)(1), the district court must conduct a hearing under section 2254(e) to determine if these facts are true.
IV
Jordan was caught between the horns of a dilemma: he was denied the right to represent himself, and then the court-appointed counsel he was forced to accept was (he says) sub-par. We Aefikm the district court’s denial of Jordan’s petition for habeas corpus on his self-representation claim. We ReveRSE and Remand on J ordan’s ineffective-assistance-of-counsel claim. We instruct the district court to hold a hearing under 28 U.S.C. § 2254(e)(2) to allow the parties to present evidence about whether Bohach had a strategic reason for failing to object to the prosecution’s improper vouching for the witness’s credibility.