Potentially equal complexity classes without known contradictory relativizations What are some examples of pairs of complexity classes $A$ and $B$ such that


we do not know whether $A=B$, and
we do not know contradictory relativizations either (i.e., we do not know oracles $P$ and $Q$ such that $A^P = B^P$ and $A^Q \ne B^Q$)?


To phrase the question another way, what are some exceptions to the heuristic that if can't figure out contradictory relativizations then it is easy to resolve the equality question outright?
I think the biggest such example at present is $BQP $ (quantum polybomial time) vs $PH $ (the polynomial time hierarchy). Significant effort has been put into separating them relative to an oracle, with no success. (Of course a powerful enough oracle will make them equal.) And the best known containment result is that $BQP $ is in $PP $. 

Some references for attacks on the oracle problem:
http://arxiv.org/abs/0910.4698
http://arxiv.org/abs/1007.0305