Relation between ExpTime vs Pspace

Question

Are there any EXPTIME-COMPLETE problems that cannot be proven to be PSPACE-COMPLETE?

Answer 1

No one knows. While it is conjectured that PSPACE $\ne$ EXPTIME, no one has any proof. In other words, it is consistent with all of our state of knowledge that PSPACE = EXPTIME. In particular:

1. If PSPACE = EXPTIME, then every EXPTIME-complete problem is also PSPACE-complete.

2. If PSPACE $\ne$ EXPTIME, then every EXPTIME-complete problem is not PSPACE-complete.

No one knows which of those two cases is true.

See, e.g., Why do we believe that PSPACE ≠ EXPTIME?.