Let me state the problem again:
Suppose PSPACE $\ne$ EXP. Is there a language in EXP - PSPACE that is not PSPACE-hard?
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.