Problems in $\text{PSPACE} \cap \text{Co-NP-Hard}$ I'm in search for examples of decision problems lying in $\text{PSPACE}\cap \text{coNP-hard}$ which are also not (known to be) in $\text{coNP}\cup \text{NP}\cup \text{NP-hard}\cup \text{PSPACE-complete}$.

Is it known any such a problem ? 

(It would be very nice to have examples of $\text{PSPACE}\cap\text{coNP-hard}$ combinatorial games with these properties, but any other kind of example would be high appreciated)
Formula Isomorphism is in $\mathsf{\Sigma_2 P} \subseteq \mathsf{PSPACE}$, is easily seen to be $\mathsf{coNP}$-hard, but is not known to be $\mathsf{NP}$-hard. Note that FI is not $\mathsf{\Sigma_2 P}$-complete unless $\mathsf{PH}$ collapses to the third level. All of this can be found in Agrawal &amp; Theirauf.