Benson–Carlson conjecture on free sphere-product actions

For a finite group $G$, define

$$h(G)=\min\{s\mid G\text{ acts freely on }\mathbb{S}^{n_1}\times\cdots\times\mathbb{S}^{n_s}\},$$

and

$$r(G)=\max\{t\mid (\mathbb{Z}/p)^t\leq G\text{ for some prime }p\}.$$

Benson–Carlson conjecture. One has $h(G)=r(G)$ for each finite group $G$. This conjecture relates the minimum number of sphere factors needed for a free action to the largest elementary abelian subgroup rank of the group. The source presents it as an extension of Smith's theorem to products of spheres; no resolution is supplied here.