Second-countable, analytic, completely Baire spaces

Question

First a few definitions. A topological space $(X,\tau)$ is said to be:

• a Polish space if it is separable and completely metrizable.
• Analytic, if there exists a surjective continuous map from a (Borel subset of) Polish space to the space $X$.
• Completely Baire (some say "hereditarily Baire"), if each closed subspace is Baire

Now my question is: do we have a counterexample of a non-Polish space that is analytic, second-countable, and completely Baire? Can we say at least that the existence of such counterexample is consistent with ZFC?

Thanks

Answer 1

This paper of Medini and Zdomskyy provides an answer: https://arxiv.org/pdf/1405.7899.pdf

They prove that you can't have a counterexample of a coanalytic separable metric space, and that this extends to projective spaces under projective determinacy. They also prove that it is consistent with ZFC that there is an analytic counterexample.