Van Engelen's Borel classification conjecture for zero-dimensional spaces

Let $X$ be a zero-dimensional Borel space that is not locally compact. A meager topological group is a topological group whose underlying space is meager, and a filter is the corresponding topological space notion used in the paper. Theorem van Engelen asserts that, for zero-dimensional $\mathbf{\Delta}^0_3$ spaces that are not locally compact, these two conditions are equivalent: being a meager topological group and being homeomorphic to a filter.

Van Engelen's conjecture. The same equivalence holds for every zero-dimensional Borel space that is not locally compact.

This extends van Engelen's theorem from zero-dimensional $\mathbf{\Delta}^0_3$ spaces to all zero-dimensional Borel spaces. The source explicitly describes the Borel version as open.