Let $X$ and $Y$ be Polish (i.e. Borel subsets of separable completely metrizable) spaces. For a Polish space $Z$, let $\mathscr{S}(Z)$ denote the limit $\sigma$-algebra on $Z$, i.e. the smallest $\sigma$-algebra on $Z$ that contains the Borel subsets of $Z$ and is closed under the Suslin operation.

**Question:** Does $\mathscr{S}(X) \otimes \mathscr{S}(Y) = \mathscr{S}(X \times Y)$?