Given PSPACE $\ne$ EXP, is there a non-PSPACE-hard language in EXP - PSPACE? Let me state the problem again:


  Suppose PSPACE $\ne$ EXP. Is there a language in EXP - PSPACE that is not PSPACE-hard?


Context

I have a problem that's in EXP. Currently I don't think it's in PSPACE. Besides proving it's EXP-complete, what else can I do? Since it seem not to be in PSPACE, I start to think it's PSPACE-hard. But is this necessary? That's why I asked this question.

Actually I have a related question, that is, given PSPACE $\ne$ EXP, whether there is a language in EXP - PSPACE that is not EXP-hard. A yes answer to the main question will imply a yes answer to this question. I think answering this question will also help me somehow.
The proof of Ladner's theorem doesn't use any special properties of P and NP and the same proof unchanged will show, assuming EXP&lt;>PSPACE, there is a language L in EXP-PSPACE and not EXP-complete under either P-time or PSPACE-reductions.

You need the full Landner look-back trick to keep L in EXP.