Mine goes unrolled under a bed.
Stepping back to my previous life as a polyurethane foam development technologist, I once tried to make foam to cascade designs spec. They specified a very low compression set which was very hard to meet.
If you leave foam very squashed for a long time, it will not totally recover, moisture will accelerate this degradation. Foam does get softer under constant flexing.
That said the foam in these mats is not there to provide padding, it is there to produce the self inflating effect and to provide at medium to damp air movements which adds comfort and insulation.