image or projection of a universally measurable set

Question

Suppose $A \subset [0,1]\times[0,1]$ is universally measurable. Is it true that its projection to the first coordinate is a universally measurable subset of $[0,1]$?

What is known is that the projection of any analytic subset (and hence also any Borel-measurable subset) of $[0,1]\times[0,1]$ is analytic and hence universally measurable, which is proved by first showing that (Lusin's theorem) the collection of universally measurable sets is closed under Suslin operations, but that doesn't seem to give any hint as to what can be said about projections of arbitrary universally measurable sets.

Answer 1

This is consistently false: Under $V = L$, there are $\Delta^1_2$ non measurable sets of reals. I don't see how to construct a ZFC counterexample though.