A Tychonoff space $X$ is called strongly zero-dimensional if each functionally closed subset $F$ of $X$ is a $C$-set, which means that $F$ is the intersection of a sequences of clopen sets in $X$.

A Tychonoff space $X$ is called almost strongly zero-dimensional if each functionally closed subset of $X$ is the union of a sequence of $C$-sets.

Question. Does there exists a (metrizable separable) Tychonoff space which is almost strongly zero-dimensional but not strongly zero-dimensional?

This problem was posed on 30.11.2019 by Olena Karlova (from Chernivtsi) on page 35 of Volume 3 of the Lviv Scottish Book.

Prize. A portrait of a mathematician who will solve this problem :)