Is there any public-key cryptosystem that is (possibly) secure against $NP\cap coNP$ adversary?
Hash based signatures come to mind; essentially, to break them, you need to find a (second) preimage of the underlying hash function, and that problem is not known to be within $NP\cap coNP$. Of course, the current instantiations are based on fixed-sized hash functions (and hence are not within $NP$); however generalizing the concept to variable length hash functions should be fairly straight-forward.