This customer has lied before on a Reg E claim and when confronted he withdrew
why are they still a customer?
is his claim an unauthorized transaction? was it a pin based transaction? did customer at any point notify the bank that he lost his debit card/credit card? i think you need to see what avenues of investigation you can undertake, which may include a conversation with him where you point out he admits on social media to being at the same place he claims an unauthorized purchase was made.
can you deny solely on the facebook post? well, you certainly have the ability to do so, but you may have a hard time defending it if a regulator comes knocking if that is your sole reason for denying the claim. I would not be comfortable denying with no additional information.