I totally agree with concerns about the thickness of the stock and have not tried to soak a T3, but..........
I did purchase the card shown below with an unusual white cloud behind the sliding runner abutting the right border. On closer inspection there was lettering from a checklist back associated with the cloud suggesting a WST. I thought the cloud could be actual paper from the wet sheet transfer.
So..........I wiped the cloud (actually the whole front) with a damp cloth and low and behold the cloud began to lift, not like a scrapbook page as a sheet, but it just slowly seemed to vaporize. The end result is seen in the second scan, not perfect, but a helluva a lot better.
The rest of the grime and sludge on the front of the card was essentially unchanged. I would expect that physically attached water based glue residue might be amenable to similar removal with a damp cloth.