Collapse versus free deformation retraction for compact polyhedra

Let $P$ be a compact $n$-polyhedron. A free deformation retraction of $P$ onto a subspace $Q$ is a homotopy $h_t:P\to P$ with $h_0=\operatorname{id}$, $h_1$ a retraction onto $Q$, and $h_t h_s=h_{\max(s,t)}$ for all $s,t\in[0,1]$. Collapse–free-retraction conjecture. The polyhedron $P$ collapses onto an $(n-1)$-polyhedron if and only if it freely deformation retracts onto an $(n-1)$-polyhedron. The claim compares simplicial collapsibility with the stronger form of topological contractibility called free contractibility. The surrounding discussion notes that collapsibility is generally stronger than free contractibility, while the two notions agree for $2$-polyhedra; the general assertion remains open.