Converse of the separable-subspace metrizability result

Determine whether every separable metrizable space is a CT3-space.

Background

Theorem CT3-imp-second proves that every separable subspace of a CT3-space is metrizable. The authors note that the converse of this statement is unresolved: it is unknown whether separability and metrizability alone imply CT3.

References

Example \ref{omega1} shows that the converse of the first statement in Theorem \ref{CT3-imp-second} does not hold; however, we do not know whether or not the converse of the second statement holds, that is, whether every separable metrizable space is $CT_3$.

— Axioms of Continuous Separation  (2608.13086 - Yang et al., 13 Aug 2026) in Remark following Theorem CT3-imp-second

Is each dense subspace of $$ $CT_3$?

— Axioms of Continuous Separation  (2608.13086 - Yang et al., 13 Aug 2026) in Section 6, Problem