It is well known that every $\mathbf{\Pi}^1_1$-set is a union of $\aleph_1$-many Borel sets. I wonder whether it can be improved under certain reasonable set theory axioms assumption.

For example, assuming $ZFC+CH$, then it is trivially true that every set is a union of $\aleph_1$-many closed sets. But this seems heavily depends on $CH$ since if $ZFC+\neg CH+MA$, then there is a lightface $\Pi^0_2$-set which cannot be a union of $\aleph_1$-many closed sets.

So my question is: is it consistent with $ZFC+\neg CH$ that every $\mathbf{\Pi}^1_1$-set is a union of $\aleph_1$-many closed sets?