Dwayne Daniels conditionally pleaded guilty to bank robbery, reserving his right to appeal the district court's denial of his motion for a hearing under Franks v. Delaware , 438 U.S. 154, 98 S.Ct. 2674, 57 L.Ed.2d 667 (1978), to probe the credibility of the FBI agent who procured a warrant for samples of his DNA. We affirm because, although the wording of the warrant affidavit is misleading, curing the inaccuracy would not defeat probable cause. In any event, Daniels has not made a substantial preliminary showing that the misstatement was deliberate or reckless.
I. BACKGROUND
In November 2012, a man entered a bank in Waukegan, Illinois, pointed an apparent handgun at a teller, gave her a bag, and demanded money. He fled on foot with $4,765. Police interviewed three bank employees and viewed security footage. The employees described the robber as 5'3? or 5'4? tall, and they recounted him wearing some sort of Halloween mask, white gloves, blue jeans, and a blue, two-toned hooded jacket. One employee said she could see the robber's bare neck where the mask ended and thought his skin was black.
*675About 15 minutes after the robbery, Waukegan police recovered several items approximately two blocks away from the bank and passed them on to FBI examiners: a toy handgun, a Santa hat, a rubber Halloween mask, a black cap with eye holes cut into it, a blue jacket, and two white gloves. The parties now agree that the black cap was not visible from the security footage, and no witness reported seeing it. The government's theory is that the robber wore the cap under the Halloween mask, and no one disputes that the cap was found near other items that comported with the security footage and witness descriptions.
An FBI examination revealed male DNA on the black cap and one of the gloves, plus male and female DNA on the second glove. A database search returned a preliminary match between DNA from the cap and Daniels, a 5'4? black male then in the custody of Lake County Jail for an unrelated offense.
Based on this initial match, the FBI sought a warrant for a fresh buccal swab and hair sample. A Special Agent's affidavit accompanied the warrant application. Paragraph #6 of the affidavit describes a man "wearing a Halloween mask, white gloves, blue jeans, and a blue, two-toned hooded jacket" who carried "a small handgun" into the bank, robbed it, and then "ran northbound through a parking lot, and then went east on Glen Flora Avenue."
Paragraph #8 continues with the description that led to this appeal:
Approximately fifteen minutes after the robbery, officers from the Waukegan Police Department who responded to the robbery recovered several items that, according to Employees A, B, and C, and corroborated by video surveillance footage from the bank, Individual A wore or used to rob the bank. Officers recovered those items near the intersection of Glen Flora Avenue and Green Avenue, which is approximately two blocks northeast of the bank. Specifically, officers recovered a blue, two-toned hooded jacket, two white gloves, a black cap with eyeholes cut into it, a Halloween mask, and a plastic toy gun partially wrapped in tape.
The affidavit further noted a match between Daniels's DNA and DNA collected from the black cap.
A magistrate judge issued the warrant and the FBI obtained the hair sample and buccal swab. With these new samples, the government confirmed the match between Daniels and the cap-and, this time, also matched him to both gloves. An indictment followed.
Daniels then alerted the district court that "[n]ot one of the three interviewed bank employees said that a 'black cap' was 'wor[n] or used to rob the bank.' " Arguing that the affidavit implied the opposite, he moved for a hearing under Franks v. Delaware , 438 U.S. 154, 98 S.Ct. 2674, 57 L.Ed.2d 667 (1978), to probe the truthfulness of the FBI's affiant. But the court denied the motion (and reconsideration).
To be sure, the court found, the supporting affidavit inaccurately suggested that bank employees identified (and surveillance footage showed) a black cap used during the robbery. But the court found the inaccuracy unnecessary to the warrant and opined that "probable cause nevertheless exist[ed]" if the affidavit were revised to reflect the truth of the matter: that the black cap was merely found among other items that had appeared on video or in the witness accounts.
The court also found that Daniels had not made a substantial preliminary showing that the falsity was knowing or reckless, instead attributing the inaccuracy to a negligent "scrivener's error." Daniels *676pleaded guilty, reserving his right to appeal the Franks issue.
II. ANALYSIS
If a defendant makes "a substantial preliminary showing" that police knowingly or recklessly (rather than negligently) used a false or materially misleading statement to obtain a warrant, the Fourth Amendment requires a hearing to further test the affiant's veracity and determine whether any evidence must be suppressed. Franks , 438 U.S. at 155-56, 98 S.Ct. 2674. We review the denial of a Franks hearing for clear error, but we review de novo any legal determinations that factored into the district court's ruling. United States v. McMurtrey , 704 F.3d 502, 508 (7th Cir. 2013).
Because the district court found that the affidavit conveys a falsehood-a finding that neither party challenges on appeal-we limit our discussion to two questions: whether the falsehood is necessary to the finding of probable cause and, if so, whether there is evidence to suggest that the Special Agent knowingly or recklessly misled the magistrate judge.
A. Probable Cause
The role of a judge considering a defendant's motion for a Franks hearing is to remove any overt falsehood from the affidavit-or else incorporate any omitted material facts that undermine probable cause, if an omission is what rendered the affidavit misleading-and see if probable cause remains. United States v. Harris , 464 F.3d 733, 738 (7th Cir. 2006). Probable cause exists when, under the totality of the circumstances, a prudent person would find it is fairly probable that evidence of a crime is present. Florida v. Harris , 568 U.S. 237, 243-44, 133 S.Ct. 1050, 185 L.Ed.2d 61 (2013).
The parties disagree about how to frame the falsehood and excise it from the affidavit. Daniels argues that all references to the black cap must be removed; this approach, of course, would defeat probable cause for a warrant to collect more of his DNA. The government counters that only the language implying that witnesses saw (or the video revealed) a black cap should be removed.
The district court, for its part, determined that probable cause remained if the paragraph were reordered and edited as follows to reflect the precise truth:
Approximately fifteen minutes after the robbery, officers from the Waukegan Police Department who responded to the robbery recovered a blue, two-toned hooded jacket, two white gloves, a Halloween mask, and a plastic toy gun partially wrapped in tape. According to Employees A, B, and C, and corroborated by video surveillance footage from the bank, Individual A wore or used these items to rob the bank. Officers recovered these items-as well as a black cap with eyeholes cut into it, which employees did not see and video surveillance did not capture-near the intersection of Glen Flora Avenue and Green Avenue, which is approximately two blocks northeast of the bank.
Daniels, however, contends that courts evaluating a Franks motion should refrain from finessing the details in the manner of the district judge's comprehensive rewrite here, and should instead confine themselves to either deleting any misleading statements or including any material omissions that undermine probable cause. But we need not weigh in on that question, because it is clear that an alternative approach advanced on appeal by the government fits clearly within the Franks framework and leads to the same result.
*677Specifically, paragraph #8's misleading description can be winnowed by discretely eliminating the implication that the cap was worn by the bank robber:
Approximately fifteen minutes after the robbery, officers from the Waukegan Police Department who responded to the robbery recovered several itemsthat, according to Employees A, B, and C, and corroborated by video surveillance footage from the bank, Individual A wore or used to rob the bank. Officers recovered thoseitems near the intersection of Glen Flora Avenue and Green Avenue, which is approximately two blocks northeast of the bank. Specifically, officers recovered a blue, two-toned hooded jacket, two white gloves, a black cap with eyeholes cut into it, a Halloween mask, and a plastic toy gun partially wrapped in tape.
This information, combined with paragraph #6's true statements of the direction of the robber's flight, of what bank employees saw, and of what the video showed, is enough for a prudent observer to infer a fair probability that the black cap was linked to the crime. And the robber's height and skin color, found in paragraph #7 of the affidavit, add at least a little bit to probable cause.
Daniels asserts that this rendering of the affidavit eliminates probable cause. Not so; the showing of probable cause is somewhat diminished, but not appreciably.
We are not persuaded by Daniels's argument that the court should have removed all references to the black cap. That the police found the cap along the robber's flight path is not false. What the affidavit misconveyed was a suggestion that employees stated (and video corroborated) that the black cap was visible during the robbery. Removing that suggestion does not defeat probable cause.
B. Affiant's State of Mind
For the sake of completeness, we note that Daniels has not made a substantial preliminary showing that the Special Agent's misstep was reckless or deliberate. To secure a Franks hearing, a defendant must put forth "an offer of proof" that is "more than conclusory" and gestures toward more than negligent mistakes. 438 U.S. at 171, 98 S.Ct. 2674. What is needed is "direct evidence of the affiant's state of mind" or else "circumstantial evidence" of "a subjective intent to deceive." United States v. Glover , 755 F.3d 811, 820 (7th Cir. 2014). Although "the defendant need not overcome the court's speculation regarding an innocent explanation," id. , the defendant's task remains "relatively difficult," McMurtrey , 704 F.3d at 509.
In the district court, Daniels flatly asserted that an inference of recklessness or worse "is inescapable." On appeal he is more specific: the government's motive to overstate the connection between the black cap and the robbery is obvious; the subsequently-drafted criminal-complaint affidavit does not contain the same misrepresentation; and the affiant removed the Santa hat from the list of items recovered, thus illustrating his overall deliberateness. But these specific contentions were not made in the district court. In any event, given the slight gap between what the affidavit said and what it should have said, a reasonable observer would find it difficult to infer anything worse than negligence. The district court did not err.
III. CONCLUSION
For the foregoing reasons, the district court's judgment is AFFIRMED.