IME, left items are pretty much lost items.
I forgot a cellular modem card on the night stand by mistake (of course). I was four hours away when I realized it. I called the property and they said the room and been cleaned and nothing was turned in. I insisted they send someone to the room while I held. Surprize! They found the item right where I left it. They FedEx'd it the next day to where I was going to be staying and all was well.
They may have cleaned the room and this doesn't say much for their cleaners. I suspect the FDC just made up the story to get me to go away, that didn't work. If the GM is not willing to step up, your only choice is to see if it would be covered by your insurance (most do).
Good luck.