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 & C.":
- The complexity of 1x1 Rush Hour  (PSPACE-complete for blocks of size 2x1 );
- [SOLVED] The complexity of planar Subway Shuffle  (PSPACE-complete even for planar graphs, a draft of the paper can be downloaded here );
- The complexity of Lunar-Lockout without fixed blocks  (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&A on cstheory).
If you have many, group them by topic.
 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