Potentially equal complexity classes without known contradictory relativizations
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