If the image was displayed from your server several years ago , and for less than one year, then it might be passed the statute of limitations already.
The rule is: for an (alleged) infringement of a copyright, the copyright owner has a claim against you for 3 years since it happened. In the best case for them, count those 3 years from the moment you took down the image from public view.
If the image was taken down from public view more than 3 years ago, then they have pretty much no chance against you (imo). Because they must have found it with a bot scanning the site, and if it was entirely private not only they couldn't find it, but also there's no provable gain you make from it, nor harm to them.
PS: make sure you do remove it completely either way.
(quick writing on mobile)
FALSE...the statute runs 3 years from "date of discovery"..always use the date of the first demand letter to guage the SOL...
You're correct, from discovery. I didn't mean to imply it is not, but to say that if the description of the OP is accurate, that is, if the image was publicly available several years ago, and for less than one year - then it wasn't available anymore from his/her server -, then it couldn't have been discovered now
I remember server logs with picscout and such bots, that parse directories, sometimes from an address that discloses that there's a "case number" associated with the image, and sometimes without. Sometimes Getty immediately sent letter, sometimes - IIRC - they did not send anything for a while, despite that logs show that picscout read the image, and/or the directory where it was.
If it was me, I would not go for the date they send letter, but from the last date they must have had the information.
That said, your guess is likely correct. I trust your experience in these cases tells you that the OP probably says "private" but means publicly accessible just not on their web pages. In that case, indeed picscout might have seen the image recently, somewhere in a forgotten directory that nobody but bots care about. In that case however, it is likely the OP doesn't profit from it, doesn't use it for financial gain, heck it's forgotten in a corner. I think they can use that to understand and negotiate low damages if any.