Fremlin's two-to-one compact-image conjecture

Let $X$ be a compact topological space, and suppose every closed subset of $X$ is a $G_\delta$ set, meaning a countable intersection of open subsets of $X$. Fremlin's two-to-one compact-image conjecture. There is an at-most-two-to-one map from $X$ into a compact metric space.

This concerns the extent to which compact spaces with the stated descriptive property are controlled by compact metrizable spaces; the supplied material gives no resolution.