The Borsuk-Ulam-type conjecture of Baum, Dąbrowski and Hajac states that given a free action of a compact group $G$ on a compact space $X$, there are no $G$-equivariant maps $X\ast G\to X$ (with $\ast$ denoting the topological join). We prove this conjecture for locally trivial principal $G$-bundles which amounts to the non-existence of $G$-equivariant maps $G^{\ast(n+1)}\to G^{\ast n}$, which in turn is a slight strengthening of an unpublished result of M. Bestvina and R. Edwards. Moreover, we show that the Baum-Dąbrowski-Hajac conjecture partially settles a conjecture of Ageev which implies the weak version of the Hilbert-Smith conjecture stating that no infinite compact zero-dimensional group can act freely on a manifold such that the orbit space is finite-dimensional.