A regular topological space $X$ is called

$\bullet$ cosmic if $X$ is a continuous image of a separable metrizable space;

$\bullet$ cometrizable if $X$ admits a weaker metrizable topology such that each point $x\in X$ has a (not necessarily open) neighborhood base consisting of metrically closed sets.

In 1989 Gruenhage proved that under PFA a cometrizable space is cosmic if and only if it contains no uncountable discrete subspace and no uncountable subspace of the Sorgenfrey line.

Can the cometrizability be moved to the right-hand part of this characterziation?

Question. Is each cosmic space cometrizable?