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I wouldn't fill out a fraud affadavit unless I was 100% sure it was fraud.
Yes, they will most likely continue to say it is yours.
And yes, they may have documentation upon which they can legitimately rely upon in verifying their determination that it appears to have been authorized by you.
The issue of factual identity theft is difficult to resolve, so the FCRA makes provision for at least obtaining blockage of the information from a consumer's credit report without resolution of the acutal issue.
The entire purpose of the FCRA identity theft process is to totally avoid any showings of actual proof by either party, and avoid the impossiblitlity of proving a negative.
It is your ASSERTION that you did not authorize the account that is the basis for obtaining CR blockage. What the creditor asserts is immaterial. They are not involved in the CR blockage process under the FCRA.
In order to prevent spurious assertions of potential identity theft, the check and balance within the FCRA is that the consumer must place that assertion into the form of a sworn statement before a law enforecement agency, thus subjecting themselves to potential criminal prosecution for any knowingly false statements.
You must include an "identity theft report," which is defined under the FCRA as a sworn statement before a law enforcement agency, and an additional statement that you never authorized the account/transaction. One CRA apparently interprets that additional statement to be in the form of an FTC Affidavit, which the FTC has posted for consumer use.
So file a police report, execute an FTC affidavit, and send it to the CRA along with proof of your identity.
That requires the CRA to immediately block that information from your CR. It is not a legal determination of actual identity thett, so dont be deterred by fear that it may later be determined otherwise. Just be full and truthful in your assertions.