Is each $G_\delta$-measurable map $\sigma$-continuous?

Question

Definition. A function $f:X\to Y$ between topological spaces is called

$\bullet$ $G_\delta$-measurable if for each open set $U\subset Y$ the preimage $f^{-1}(U)$ is of type $G_\delta$ in $X$;

$\bullet$ $\sigma$-continuous if $X$ has a countable cover $\mathcal C$ such that $f|C$ is continuous for every $C\in\mathcal C$;

$\bullet$ piecewise continuous if $X$ has a countable closed cover $\mathcal C$ such that $f|C$ is continuous for every $C\in\mathcal C$.

By an old result of Jayne and Rogers, each $G_\delta$-measurable maps $f:X\to Y$ between analytic spaces is piecewise continuous and hence $\sigma$-continuous.

Question. Is each $G_\delta$-measurable function between separable metrizable spaces $\sigma$-continuous?

Answer 1

I looked to my own old paper with Bokalo and have found there Example 9.3 answering this problem:

Example. Under Martin's Axiom (more precisely, $\mathrm{add}(\mathcal M)=\mathrm{cof}(\mathcal M)$) there exists a bijective function $f:X\to Y$ between zero-dimensional separable metrizable spaces such that $f^{-1}$ is continuous, $f$ is $G_\delta$-measurable but $f$ is not $\sigma$-continuous.

In fact, the problem was motivated by this question of Karlova, which is still open.