# Does the pointclass of universally Baire sets always have the uniformization property?

A set of reals, or binary relation on the reals, etc., is called universally Baire if and only if every continuous preimage of it in every topological space has the property of Baire. (There is also a useful characterization involving absolutely complementing pairs of trees.)

A pointclass $\Gamma$ is said to have the uniformization property if and only if for every binary relation $R \in \Gamma$ there is a function $f \in \Gamma$ such that $f \subset R$ and the domain of $f$ equals the domain of $R$. (Here we identify a function with its graph.)

Steel proved that if there is a proper class of Woodin cardinals, then the pointclass of universally Baire sets has the uniformization property.

Is the conclusion of Steel's theorem provable in ZFC? I would guess not, but I have not been able to come up with a counterexample. In particular, I do not know whether or not it holds in $L$.