The minimal-sphere-product existence problem for finite group actions

Let $G$ be a finite group, and let $r(G)$ denote its rank, namely the maximum rank of an elementary abelian $p$-subgroup of $G$ over all primes $p$ dividing $|G|$. Minimal-sphere-product conjecture. There should exist a finite complex $X$ with a free $G$-action such that $X$ has the homotopy type of a product of $r(G)$ spheres. This is the natural extension of Swan's rank-one theorem, which gives a finite complex homotopy equivalent to a sphere when $r(G)=1$. The source describes the problem as difficult and open; constructing such minimal free actions is a central problem for groups without periodic cohomology.