Arkansas inmate Mikel W. Houston appeals the district court’s denial of his petition for a writ of habeas corpus without an evidentiary hearing. Houston claims that, because of the ineffective assistance of his trial counsel, favorable polygraph test re-*1248suits were not admitted into evidence in the jury trial that led to his conviction for rape of his twelve-year-old daughter. On the unique facts of this case, we conclude that Houston is entitled to an evidentiary hearing on his ineffective assistance claim. Accordingly, we reverse.
I.
In December 1985, Houston’s twelve-year-old daughter, L, told North Little Rock police that he had digitally raped her on two occasions that fall. Houston was subsequently charged with rape by deviate sexual activity. On October 1, 1986, six weeks before trial, Houston underwent a polygraph exam at his own expense. Twice, he answered a series of six questions:
1. Do you intend to answer each question on this test truthfully?
A. Yes.
2. Have you ever with sexual intent put your hand on [L’s] genital area?
A. No.
3. Have you ever put your finger in [L’s] vagina?
A. No.
4. Have you ever with sexual intent fondled [L] in any manner?
A. No.
5. Have you invented or made up any part of your story that you told me today concerning these allegations?
A. No.
6. Have you deliberately lied to any question I have asked you on this examination?
A. No.
On October 7, the polygraph examiner sent the test results to Houston’s trial attorney along with a report stating:
Analysis of Mr. Houston’s polygrams revealed no attempted deception to the relevant questions. It is my opinion he has been truthful in his answers during the polygraph examination.
Although this report and the polygraph test results are part of the record of Houston’s postconviction proceedings, they appear nowhere in the trial record. They were not mentioned at any recorded pretrial proceeding nor offered into evidence at trial, and this omission was not raised as an issue on direct appeal.
Houston was tried on December 9, 1986. The trial testimony established that Houston and Linda Teas divorced in 1977; L was their only child. As part of an established visitation schedule, L spent most weekends in 1985 at the two-bedroom house where Houston lived with his mother, his brother and sister-in-law, and their two young children. When L visited, she and at least one other child shared a queen-size bed in the bedroom where Houston’s semi-invalid mother slept in a special hospital-style bed. Houston’s brother and sister-in-law slept in the second bedroom, and Houston slept on a couch.
Through the summer of 1985, L was excited about these visits. In the fall, however, Houston twice scolded L, first when she drove a moped across several city blocks without permission, and on another visit when she picked up a loaded gun. After the moped incident, Houston immediately took L home and told her she could “stay there until you learn to mind me”; after the gun incident, Houston “cautioned” her.
In November, L told one of her cousins that her father had “touched” her. The cousin told Houston’s sister-in-law, who questioned L. When pressed for details, L said, “I knew you wouldn’t believe me,” and ran from the room. L’s mother heard the story in early December and took L to the North Little Rock police, who took a taped statement. On the advice of police, Teas next went to Arkansas Children’s Hospital, where L was given a physical examination that revealed no abnormality, a result consistent, the examining physician later testified, both with digital rape and with no rape at all.
Given the lack of supporting medical evidence, L’s testimony was the prosecution’s only direct proof of the alleged rape. L testified that during the night of October 26, 1985, when she awoke from sleeping on the queen-size bed in her grandmother’s bedroom, her underpants were around her *1249knees and her father “took his finger and pushed it in and out” of her vagina. She further testified that on a later visit “the same thing” happened, but this time Houston “used something greasy." Linda Teas testified that her daughter was “not exaggerating about this.” However, Teas also admitted that she had tried to have the charges dropped but was unsuccessful — a North Little Rock detective told her that the matter was out of her hands.
Houston took the stand in his own defense. He denied ever sexually molesting his daughter and noted that L had accused him of rape shortly after he scolded her about the moped and the loaded gun. He stated that he had only gotten into bed with L and the other children on one occasion, when he came in from work around midnight, dozed on the couch, woke up cold, could find no extra blankets, and went into the bedroom where the children were sleeping. By picking up and moving each child, he was able to lie down fully clothed on the outer portion of the mattress and partially cover himself with a blanket. L was the closest child to him.
Houston’s testimony was partially corroborated. L’s eleven-year-old cousin testified that, on the night Houston got into bed with the children, the cousin stayed awake after Houston moved him and saw nothing happen between Houston and L. Houston’s sister-in-law testified that she sat in the dining room reading that night, could see virtually the entire bed where Houston and the children lay, and did not see Houston rape his daughter. She further testified that about a week before L complained about Houston, L had accused Houston’s brother of lying on a bed “touching himself” and looking at her; unbeknownst to L, the sister-in-law had been in the bedroom at the time and knew the accusation to be false.
The jury returned a guilty verdict and, after a bifurcated penalty proceeding, sentenced Houston to the maximum penalty, life in prison.1 On appeal to the Arkansas Supreme Court, Houston unsuccessfully challenged only the sufficiency of the evidence and the fairness of the guilt/punishment bifurcation procedure. See Houston v. State, 293 Ark. 492, 739 S.W.2d 154 (Ark.1987).
Houston then filed a pro se petition seeking postconviction relief under Arkansas Criminal Procedure Rule 37. He attached the polygraph examiner’s report and alleged ineffective assistance of his trial counsel because:
Both defense and prosecution agreed to use polygraph results at the trial, but after the petitioner’s polygraph was administered twice, and showed the petitioner to be telling the truth, the state decided against the polygraph results, and defense counsel made no adverse issue , of it.
The Arkansas Supreme Court summarily denied the motion, concluding that Houston could satisfy neither prong of the ineffective assistance of counsel standard adopted in Strickland v. Washington, 466 U.S. 668, 104 S.Ct. 2052, 80 L.Ed.2d 674 (1984). The polygraph test results were inadmissible because there was no evidence of a written agreement by both parties agreeing to their admissibility, the Court explained, citing Foster v. State, 285 Ark. 363, 687 S.W.2d 829, 832 (1985), cert. denied, 482 U.S. 929, 107 S.Ct. 3213, 96 L.Ed.2d 700 (1987). Therefore, counsel’s performance was not deficient because “an objection by the defense would have been futile.”2 Moreover, Houston had failed to prove the requisite prejudice because “[w]e cannot say that there is a reasonable probability that the test results would have so bolstered the petitioner’s credibility that it is likely he would have been acquitted.” Houston v. State, No. CR 87-85, 1988 WL *125014162, 1988 Ark. LEXIS 121 (Ark. Feb. 22, 1988).
Houston then filed this pro se habeas corpus petition in the district court, alleging that his trial counsel had provided ineffective assistance because:
defense attorneys were ineffective when they failed to make an issue of the state’s breach of agreement to allow polygraph results of the petitioner to be presented as evidence____
In an in camera hearing the prosecutor, defense and the judge3 had a verbal agreement that if the petitioner would submit to a polygraph examination, that the results of that test, whether positive or negative, would be permitted to be used as evidence.
However, after ... said tests, the prosecutor reneged on the agreement, stating “polygraphs can only hurt my case”____ [T]he defense failed to make an issue of the matter, and neglected to raise the matter on direct appeal to the Arkansas Supreme Court.
The district court denied Houston’s petition without a hearing, agreeing with the Arkansas Supreme Court that “an objection by counsel would have been futile” because the polygraph tests were inadmissible absent a written agreement between the parties. Houston v. Lockhart, No. PB-C-88-294 (E.D.Ark. Aug. 28, 1990). On appeal, after we appointed counsel and heard oral argument, a panel of this court affirmed, one judge dissenting. We subsequently vacated the panel opinion and granted rehearing en banc. Houston v. Lockhart, 958 F.2d 826 (8th Cir.1992). We now reverse and remand for an evidentiary hearing.
II.
“Under 28 U.S.C. § 2254, a federal district court must hold an evidentiary hearing when a [habeas] petition alleges sufficient grounds for release, relevant facts are in dispute, and the state courts did not hold a full and fair evidentiary hearing.” Spillers v. Lockhart, 802 F.2d 1007, 1009 (8th Cir.1986). However, a petition may be summarily dismissed if its factual allegations are “palpably incredible” or “patently frivolous or false.” See Blackledge v. Allison, 431 U.S. 63, 75-76, 97 S.Ct. 1621, 1629-30, 52 L.Ed.2d 136 (1977).
In this case, Houston has alleged that he took two polygraph tests after counsel orally agreed the test results would be admissible, that the prosecutor then reneged on her oral agreement because the test results were favorable to Houston, and that defense counsel made no attempt to offer the favorable test results into evidence or to raise the issue on direct appeal. Those allegations are not admitted by the State, are not palpably incredible, and were never the subject of a hearing in state court. Therefore, Houston is entitled to an evidentiary hearing in federal court if those allegations contain sufficient grounds for habeas corpus relief under Strickland.
III.
In the final analysis, this case turns primarily on the alleged oral agreement and the conduct of Houston’s trial counsel before trial. But we emphasize at the outset that it is the trial that matters. The jury that convicted Houston never learned that he had twice passed a polygraph test, and the appellate court that affirmed his conviction never learned that such evidence existed but was not admitted at trial. Indeed, the test results were never offered into evidence. Although the parties imply that defense counsel knew before trial the court would exclude the test results if offered, the record is totally silent on the subject.
This case is unusual because we are dealing with counsel’s failure even to offer a piece of evidence that was indisputably favorable to the defense, evidence akin to the testimony of an unimpeachable (but not necessarily credible) alibi witness. One of defense counsel’s duties is “to bring to bear such skill and knowledge as will render the trial a reliable adversarial testing process.” Strickland, 466 U.S. at 688, 104 *1251S.Ct. at 2065. To fulfill that duty, counsel must offer at trial material, admissible evidence that is favorable to the defense. Thus, as the Arkansas Supreme Court recognized, the ineffective assistance issue in this case turns on two questions: (1) whether the polygraph test results were admissible, or so arguably admissible that counsel’s failure to offer them fell below an objective standard of reasonable attorney performance, and (2) whether counsel’s failure mattered, that is, whether there is a reasonable probability that the outcome of Houston’s trial would have been different had the test results been admitted into evidence. We review both of these issues de novo. See Strickland, 466 U.S. at 698, 104 S.Ct. at 2070; Garmon v. Lockhart, 938 F.2d 120, 121 (8th Cir.1991). Admissibility.
An Arkansas statute provides that polygraph test results “shall be inadmissible in all courts in this state.” Ark.Stat. Ann. § 12-12-704. However, the Arkansas Supreme Court has not interpreted this statute literally; rather, such test results “are only admissible if both parties enter into a written stipulation agreeing on their admissibility.” Foster, 687 S.W.2d at 832. Given this clear state law, we agree with the Arkansas Supreme Court and the district court on one important proposition — if defense counsel had approached the prosecutor before or during trial with unsolicited4 favorable polygraph test results, the prosecutor could have refused to enter into a written stipulation as to admissibility, the test results would then have been inadmissible, and any attempt to offer them at trial would have been futile.
However, in this case Houston alleges that defense counsel and the prosecutor orally agreed, before the polygraph test was taken, that the test results would be admissible. The Arkansas Supreme Court and the district court held that defense counsel could not be guilty of ineffective assistance because the prosecutor had no duty to reduce this agreement to writing. We disagree. Assuming the truth of Houston’s allegations of an oral agreement, as we must at this stage of the case, we conclude that defense counsel’s performance fell below the objective standard of reasonableness both before and after Houston took the polygraph test.
(1) Before Houston took the polygraph test, defense counsel should have asked the prosecutor to put the alleged agreement in writing, or on the record in the presence of the trial judge. The statute and prior Arkansas cases made it clear that an oral agreement would be unenforceable. See State v. Bullock, 262 Ark. 394, 557 S.W.2d 193 (1977). In this legal environment, failure to get an oral agreement reduced to writing “betray[s] a startling ignorance of the law” that is “contrary to prevailing professional norms.” Kimmelman v. Morrison, 477 U.S. 365, 385, 106 S.Ct. 2574, 2588, 91 L.Ed.2d 305 (1986). Compare Betancourt v. Willis, 814 F.2d 1546 (11th Cir.1987).
Nor can we assume, before Houston took the test, that it would have been futile for defense counsel to ask the prosecutor to put their oral agreement in writing. The prosecutor probably would have signed a written stipulation because she wanted to obtain admissible polygraph test results. At a minimum, having made an oral agreement, it is likely she would have reduced the agreement to writing because that is the ethical thing to do.5 Moreover, Hous*1252ton has alleged that the trial court was a party to the oral agreement; in that situation, had defense counsel pointed out the problem, the court would likely have placed the agreement on the record to satisfy the Arkansas Supreme Court’s requirement of a “written stipulation.” Houston deserves an evidentiary hearing to explore these issues.
(2) After Houston completed the polygraph tests based only upon an oral agreement, the dynamics of the situation changed. Since the test results were favorable to Houston, the prosecutor had presumably lost interest in stipulating to their admissibility. Even so, we conclude that defense counsel should have requested that the prosecutor sign such a stipulation and, if the prosecutor declined to do so (as Houston alleges), offered the test results into evidence without the stipulation.
In the first place, if faced with the question whether to renege on her oral agreement on the record, the prosecutor might have relented and signed a written stipulation. Second, the trial court might have decided to enforce the oral agreement by admitting the test results into evidence, either as a matter of public policy,6 or on the authority of Whiteside v. State, note 4 supra. Finally, even if the prosecutor refused to sign a stipulation and the trial court excluded the evidence, defense counsel would have made a record on a potentially powerful issue on appeal.
For all of these reasons, we conclude that, if the alleged oral agreement existed, defense counsel’s failure to make a record of that agreement and attempt to get the favorable polygraph test results into evidence fell below an objective standard of reasonable attorney performance. Thus, Houston’s allegations warrant an evidentiary hearing under the first prong of the Strickland test.
Prejudice.
To satisfy the second prong of Strickland, Houston must prove that the outcome of his trial “would reasonably likely have been different absent [defense counsel’s] errors.” Strickland, 466 U.S. at 696, 104 S.Ct. at 2069.7 This requires us to weigh the likely impact on Houston’s trial if the polygraph test results had been admitted into evidence. The prosecution's case was based upon L’s testimony that Houston had digitally raped her. That testimony was directly contradicted by Houston, whose version of the events was partially corroborated by two eyewitnesses. The jury was confronted with two colliding versions of the truth; it had to decide whether to believe the father or the daughter. Would polygraph test results supporting Houston’s denial have mattered?
In McMorris v. Israel, 643 F.2d 458, 462 (7th Cir.1981), cert. denied, 455 U.S. 967, 102 S.Ct. 1479, 71 L.Ed.2d 684 (1982), the Seventh Circuit held that favorable polygraph test results are “materially exculpatory” for due process purposes in a case where the defendant’s credibility is crucial. The court explained:
The entire prosecution case rested on the testimony of the victim____ The defendant took the stand and denied any involvement in the crime. His credibility was the primary issue before the jury, and the jury’s verdict reflects its decision not to credit his testimony. Under these circumstances, a polygraph examination that bolstered the defendant’s credibility might easily have been decisive in securing an acquittal.
*1253Similarly, the Third Circuit has held that the improper withholding of polygraph test results that damaged the credibility, of a key prosecution witness was a “material” violation of Brady v. Maryland, 373 U.S. 83, 83 S.Ct. 1194, 10 L.Ed.2d 215 (1963), because “there is a reasonable probability that the result of the trial would have been different had the prosecution properly disclosed” this evidence. Carter v. Rafferty, 826 F.2d 1299, 1309 (3rd Cir.1987) (citation omitted).
In our view, these decisions only confirm the obvious — that polygraph test results may well have a significant influence on the jury in a case that turns on the credibility of two conflicting witnesses. Indeed, the concern, that juries will give too much weight to polygraph evidence is one reason it has historically been inadmissible in Arkansas. See Bullock, 557 S.W.2d at 194. The assumption that such evidence has potent force in a jury trial is illustrated by Wingfield v. State, 303 Ark. 291, 796 S.W.2d 574, 576-77 (1990), where the court held that a mistrial was the only appropriate remedy for the prejudice introduced by two improper references to an inadmissible polygraph examination. See also United States v. Alexander, 526 F.2d 161, 168 (8th Cir.1975) (discussing the “potentially pervasive influence of polygraph evidence upon the jury”).
Based upon these authorities and our own review of the trial record, we conclude that, if the polygraph test results were, to a significant likelihood; admissible, Houston was prejudiced by his counsel’s failure either to get them admitted into evidence or to make a record of his efforts to do so for purposes of appeal.
For the foregoing reasons, we remand this case for an evidentiary hearing into whether Houston received the ineffective assistance of his trial counsel as a result of the favorable polygraph test results being neither offered nor admitted into evidence.
. The Court's reference to an objection is troubling. Because the polygraph test results were favorable, it was defense counsel’s burden to make an affirmative effort, either before or during trial, to get the test results into evidence, or at least to make a record of his effort to do so. Thus, the ineffective assistance question in this case has nothing to do with objections.
. This is the first time Houston alleged that the trial judge was a party to the oral agreement. This allegation is not critical to our conclusion that he is entitled to an evidentiary hearing.
.This is an important caveat. In Whiteside v. State, 12 Ark.App. 271, 675 S.W.2d 645 (Ct.App.1984), the court held that it was not error to admit unfavorable polygraph test results because defendant had orally agreed with the prosecutor and the trial judge that, if he was permitted to take a polygraph test, the results would be admissible. Whiteside suggests that the polygraph test results might be admissible under Arkansas law if Houston took the tests in reliance upon the prosecutor’s assurance that the results would be admissible.
.For example, the Seventh Circuit’s new Standards for Professional Conduct include among the Lawyers’ Duties to Other Counsel:
6.We will adhere to all express promises and to agreements with other counsel, whether oral or in writing, and will adhere in good faith to all agreements implied by the circumstances or local customs.
. See United States v. Shapiro, 879 F.2d 468, 471 (9th Cir.1989) ("This court ... enforce[s] agreements made by prosecutors upon which defendants have justifiably relied to their detriment” ); Rowe v. Griffin, 676 F.2d 524, 536 n. 3 (11th Cir.1982) (“under appropriate circumstances informal [prosecutor] promises may be enforced”); United States v. Hudson, 609 F.2d 1326, 1328 (9th Cir.1979) (recognizing "the responsibility of federal prosecutors meticulously to fulfill their promises”); Workman v. Commonwealth, 580 S.W.2d 206, 207 (Ky.1979) (enforcing agreement to dismiss indictment if defendant passed a polygraph test); State v. Davis, 188 So.2d 24 (Fla.App.1966) (same).
. This standard is "somewhat lower” than the "outcome-determinative standard ... for assessing motions for new trial based on newly discovered evidence.” Id. 466 U.S. at 693-94, 104 S.Ct. at 2068.