Sometimes I've kept my self-inflating sleeping pads stored flat, unsealed (hence inflated) on top of the gear boxes so there is no weight on them.
Other times, I've kept the pads in the closet on a pants-hanger clamped on the thin, sealed edge. Again, I store them with the valve open.
I've had a lot of self-inflating pads in my life and I rented out dozens while working in a BP store. One advantage of storing them non-compressed between trips is that they self-inflate faster on a trip. I find the difference most marked when they are cold and slow to inflate.
When I'm really on top of my game, I try to get them dried out when I can, which is most easily done in winter. Let them fill with cold (therefore dry) outside air, bring it inside to warm up to some moderately high temp (100F-ish) and then squeeze out the air that now contains some of the moisture that you'd introduced when inflating or topping it off by mouth. I just do 2-3 times, but better yet would be weigh it before each cycle until you see no change in weight.