For what its worth, I store wet honey supers on top of my hives above an
inner cover with a 2 inch hole in the center until weather cools (Oct/Nov in
South Texas).  I have had no problems with queens moving up.  I move supers
to a pallet in the shade in the bee yard. I stack the honey supers with a
queen excluder on bottom and cover the top. I have left unused honey supers
stacked in this manner all year with no moth damage.
    For me, the advantages of no chemicals and having equipment already in
the bee yard outweigh the disadvantage of having to remove a couple of supers
when I need to work the hives during the summer.
    BA