Problems not known to be PSPACE-complete What are problems with the following properties:

1) they are restriction of (possibly well known) problems that are PSPACE-complete;

2) the restricted versions are in PSPACE, but it is an open problem if they are PSPACE-complete (or even if they are NP-hard).

Four examples from "puzzles &amp; C.":


The complexity of 1x1 Rush Hour [1] (PSPACE-complete for blocks of size 2x1 );
[SOLVED] The complexity of planar Subway Shuffle [1] (PSPACE-complete even for planar graphs, a draft of the paper can be downloaded here );
The complexity of Lunar-Lockout without fixed blocks [1] (PSPACE-complete with fixed blocks);
(not so famous) The complexity of (my) Switch-network problem (it is a restriction of the PSPACE-complete Sokoban, NP-hard in the non-planar case, see this Q&amp;A on cstheory).


If you have many, group them by topic.

[1] Robert A. Hearn, Erik D. Demaine: Games, puzzles and computation. A K Peters 2009, ISBN 978-1-56881-322-6, pp. I-IX, 1-237
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