It is well-known that (assuming P$\neq$NP), there are problems in NP that are not NP-hard neither in P. Such class of problems are called NP-intermediate (https://en.wikipedia.org/wiki/NP-intermediate), and one candidate problem that might be NP-intermediate is graph isomorphism.
I was wondering whether there is a similar notion for higher classes in the polynomial hierarchy. In particular, is there a similar notion for $\Pi_2^P$? That is, something like "the class of problems in $\Pi_2^P$ that are not $\Pi_2^P$-hard but do not belong to a lower-level in the hierarhcy (should we say $\Delta_2^P$?).
If that is the case, is there a candidate problem that we suspect might belong to such class?
