Strong Choquet preimage implies strong Choquet?

Recall that a strong Choquet space is one where player II has a winning strategy in the game where two players take turns: player I chooses an open set and a point inside, then player II chooses a smaller open set containing the chosen point, then player I chooses a smaller still open set and a point inside, and so on, and player II wins if the intersection is nonempty after $\omega$ turns.

Suppose we have two compact Hausdorff spaces $X,Y$ and a continuous surjection $f\colon X\to Y$. Suppose in addition that we have $Y'\subseteq Y$ such that $f^{-1}[Y']$ is strong Choquet. Is it true, then, that $Y'$ is strong Choquet as well?

If $X,Y$ are metrisable/second countable, then this is true: strong Choquet subspaces are exactly the $G_\delta$ subsets, and it follows from e.g. exercise 24.20 in Kechris that if $f^{-1}[Y']$ is $G_\delta$, then so is $Y'$. But what about in general?

Also, if $f$ was open, this would be true (as continuous open images of strong Choquet spaces are strong Choquet), but $f$ is in general closed and not open.

If it helps, we may assume that $X,Y$ are both zero-dimensional, i.e. have bases of clopen sets.