Amex will go crazy at first and soft pull your credit reports for the first 60 days in particular. This includes all three Credit Bureaus. In essence, they are babysitting you to see if you warrant their 3x CLI. Nontheless though, unless if you've really really angered AMEX they would not hard pull, but soft pull as everyone else had said. Assuming you haven't gone on a tear asking for new cards from them or CLI's... Then that's odd. Defintely contact them. The other interesting thing regarding a hard pull is that a creditor has to give you a copy. So have you recieved a copy 10 times? What's even stranger though is why their living on your Equifax/Transunion the most. There preferences are high on Experian, then Equifax, and lastly Transunion. I would expect the exact opposite sequence as you mentioned (not questioning you) just odd.
Ultimately, an unathorized lender hard pull leaves them liable to remove the pull and 1000 dollars in court for each pull. I am also in the camp that I believe you are mistaking hard and soft pulls. Defintely worth checking out though.
