The Baire–banalytic characterization conjecture for Hausdorff topological groups

Let $X$ be a Hausdorff topological group. Baire–banalytic characterization conjecture. The group $X$ is Polish if and only if $X$ is a Baire space and a banalytic space. This conjecture would remove the assumption of PFA from the characterizations in the preceding theorems; its status is not resolved in the supplied source.