Will do. I also move mine to the cold fridge after they bloom - I keep them in individual ripening containers before that in the "cave." Once they have bloomed well, I wrap and move to the cold fridge.
I try to have them spread out in the cold fridge, but sometimes my wife thinks there ought to be other kinds of food stored in the fridge -- go figure.
As best I recall, last time I had them spread out by the time they were getting squishy. But even if they were stacked, the paper helps to hold the shape, so I would not be too worried.
"One of these days" I plan to make a little 3-tier stand that will hold my usual three-cam make in the cold fridge in a "separated stack" -- all in a vertical column, but with airspace under and between. Something like this, where the dashed lines represent the shelves, the vertical lines represent a post on the side, and the solid line on the bottom is a base:
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That will let them take up the minimum space in the fridge, but still (hopefully) provide optimal ripening conditions -- actually, even better conditions than I would have with them spread out, since then the bottom of each cheese is "covered" by contact with the shelf. (Glass shelves in my fridge, not wire.)