They claim everything stays BUT you can't open any new boxes until you have an open inventory slot.
Everything you get will go into the SHOP/Unclaimed boxes area until you free up an inventory slot. You can also buy with tokens more slots if you so choose.
YES this inventory limit does suck especially at such a low number for 3 mechs. It should be at least 300 to 500
I still have 3,000 items left even after all the fusing I've done.