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Intermediate problems between P and NP are quite famous, and are sometimes considered as complexity classes by themselves.

Do you know of any problem that is known to be PSPACE-hard and in EXPTIME, and resisting all efforts to be proved complete for one of these classes ?

Lifted (succinct) versions of problems between LOGSPACE and PTIME are accepted, but even more interesting would be problems that are not of this form.

Here are some I found, this area of complexity theory seems to be the realm of games:

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    $\begingroup$ I think the more correct way of stating the question is asking for candidate problems in ExpTime - PSpace - ExpTime-hard. Note that e.g. in the case of P vs. NP we don't know if Factoring or GI is P-hard. $\endgroup$ – Kaveh Jun 10 '17 at 18:38

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