Benson–Carlson rank conjecture for free actions on products of spheres

Let $G$ be a finite group. Define its rank by

$$\operatorname{rk}(G)=\max\\{r\mid (\mathbb{Z}_p)^r\leqslant G\text{ for some prime }p\\},$$

and define $h(G)$ to be the least integer $k$ such that $G$ acts freely on a product $S^{n_1}\times\cdots\times S^{n_k}$ of spheres. Benson–Carlson conjecture. One should have

$$\operatorname{rk}(G)=h(G).$$

This conjecture extends the elementary-abelian free-rank problem to arbitrary finite groups. The quantity $h(G)$ is well-defined by Oliver's result that every finite group acts freely on some product of spheres; the equality remains open in general.