Baum–Dąbrowski–Hajac conjecture on equivariant maps from joins

Let $X$ be a compact Hausdorff space with a continuous free action of a nontrivial compact Hausdorff group $G$. Let $X*G$ denote the join equipped with the diagonal action of $G$. Baum–Dąbrowski–Hajac conjecture. There does not exist an equivariant continuous map

$$f:X*G\to X.$$

The proposition immediately preceding this conjecture gives a partial solution when $X$ is a free $H$-space that is compact, or paracompact and finite dimensional, and $H$ is a topological group with a nontrivial finite subgroup. The general compact Hausdorff case stated here remains unresolved in the supplied text.