The Cantor-space conjecture for crowded Baire CDH groups

A topological group is CDH (countable dense homogeneous) if any two countable dense subsets are carried to one another by a homeomorphism of the group. A space is crowded if it has no isolated points, and Baire if the intersection of every countable family of dense open sets is dense. The Cantor space is $2^\omega$. Cantor-space conjecture. Every crowded Baire CDH group contains a copy of $2^\omega$.

This conjecture is intended to resolve the cited open problem about crowded Baire CDH spaces in the realm of groups. The source presents it as open and gives no resolution.