The stronger finite-space form of Webb's conjecture for elementary abelian p-subgroups

Let $G$ be a finite group and let $p$ be a prime dividing $|G|$. Let ${\mathcal A}_p(G)$ be the poset of non-trivial elementary abelian $p$-subgroups of $G$, and let ${\mathcal A}_p(G)'$ denote its poset subdivision, consisting of non-empty chains. The quotient ${\mathcal A}_p(G)'/G$ is regarded as a finite topological space.

Stronger Webb conjecture. The finite space

$${\mathcal A}_p(G)'/G$$

is contractible.

This strengthens Webb's reformulation, which only asserts that this quotient is homotopically trivial. The authors report that no non-contractible example is known and prove contractibility in particular cases, but leave the general statement conjectural.