Problems not known to be PSPACE-complete
Retrograde Chess. It is $PSPACE$-complete if you are allowed to have arbitrarily many kings and none of them can be in check at any time. If no (or only one per player) kings are allowed, it is known that there are positions that require exponential moves, but the problem is only known to be $NP$-hard.

http://arxiv.org/abs/1409.1530

https://mathoverflow.net/questions/27944/do-there-exist-chess-positions-that-require-exponentially-many-moves-to-reach