Choquet property for spaces with an H-closed π-base

Ascertain whether every Hausdorff topological space that has a π-base whose elements have H-closed closures is a Choquet space, equivalently, whether Bob has a winning strategy in the Banach–Mazur game on the space.

Background

The authors recall that if a space has a π-base whose elements have compact closures, then it is Choquet (Theorem 1.6). They ask whether the analogous statement is true when compactness is replaced by H-closedness for the closures of π-base elements. Later, they provide a partial answer by showing that a quasiregular space with an H-closed π-base is Baire (Theorem 4.5), but this does not settle the Choquet question.