Is there an almost strongly zero-dimensional space which is not strongly zero-dimensional 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 :)
 A: Not an answer, but this may be helpful:
Theorem 1. If $X$ is a Lindelöf Tychonoff almost strongly zero-dimensional space, then the  following are equivalent:
(i) $X$ is strongly zero-dimensional;
(ii) $X$ is almost zero-dimensional, that is, $X$ has a neighborhood basis of C-sets.
Proof. (i)$\Rightarrow$(ii) is trivial, and the converse follows from Theorem 4.3 in this paper (we assume separable metrizable there, but Lindelöf should be enough). $\square$
In light of Taras Banakh's comment above, for separable metrizable spaces I believe the question is: If $X$ is separable metrizable and $f:X\to Y$ is a continuous bijection onto a zero-dimensional space $Y$ which maps open sets to $G_\delta$-sets, then is $X$ almost zero-dimensional?
Theorem 2. Every almost strongly zero-dimensional homogeneous Polish space $X$ is (strongly) zero-dimensional.
Proof. If $U$ is any open subset of $X$, then $U$ is a $\sigma$C-set, so by the Baire property there is a C-set $F\subseteq U$ which contains a non-empty open set.  Continuing this process we construct C-sets $F_n$ such that $F_{n+1}\subseteq F^\mathrm{o}_n$ and $\text{diam}(F_n)\leq 1/n$ in a complete metric.  Then there exists $x\in \bigcap F_n$, and $x$ has a neighborhood basis of C-sets.  By homogeneity, $X$ is almost zero-dimensional, so by Theorem 1 $X$ is strongly zero-dimensional. $\square$
More generally it is true that each almost strongly zero-dimensional Polish space is zero-dimensional at a dense $G_\delta$-set of points.
