I do the more or less the same: after the final wash, I lay the print on a piece of plexi glass and use the squeegee to get rid of water. What is different is that I then hang the prints to dry. See the attached picture - these are 20x24in. prints (50x60cm). This is in my workspace, I generally go home after printing. The prints dry slow overnight and in the morning there are never waterdrops left.
After that I press them flat and put the prints between acid free carton, to let them rest for a day or so.
I would not put the emulsion side of the wet print in contact with anything.
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