Natural problems in $NP \cap coNP$ not in $UP \cap coUP$? Are there any natural problems in $NP \cap coNP$ that are not (known to be/thought to be) in $UP \cap coUP$?

Obviously the big one everyone knows about in $NP \cap coNP$ is the decision version of factoring (does n have a factor of size at most k), but that is in fact in $UP \cap coUP$.
While parity games are known to be in both, it's been claimed that stochastic parity games are not known to be in UP intersect coUP.