If a normal, first countable space is the union of countably many open metrisable subspaces, must that space be metrisable?

The answer is consistently yes if the space has cardinality $\omega_1$.

If a normal, first countable space is the union of countably many open metrisable subspaces, must that space be metrisable?

Partial answers, which I proved in the 1980's, include:

(0) The answer is consistently yes if the space has cardinality $\aleph_1$.

(1) Yes under MA, if the cardinality of the space is less than $\mathfrak{c}$.

(2) Yes, if the space is countably metacompact.

(3) Such a space must be collectionwise Hausdorff.

(4) It is not known if such a space is collectionwise normal (at least not to me).

(5) Any counterexample would be a Dowker space with a $\sigma$-disjoint base.

Is a normal, first countable space which is the union of countable many open metrisable subspaces normal?

The answer is consistently yes if it has cardinality omega_1

If a normal, first countable space is the union of countably many open metrisable subspaces, must that space be metrisable?

The answer is consistently yes if the space has cardinality $\omega_1$.