The rank conjecture for free actions on products of spheres

Let $G$ be a finite group, and let $G$ act freely on

$$X=\mathbb S^{n_1}\times\dots\times\mathbb S^{n_m}.$$

The rank $r(G)$ is the maximum of the $p$-ranks $r_p(G)$ over primes $p$ dividing $|G|$, where $r_p(G)$ is the largest $r$ such that $(\mathbb Z/p)^r$ is a subgroup of $G$. Rank conjecture. If $G$ acts freely on $X$, then

$$r(G)\le m.$$

This extends the rank-one restriction for free actions on homotopy spheres and asks for the minimal number of sphere factors needed for a free action. The statement is presented as an open conjecture; it is known for $m=2$ and in several equidimensional cases, but remains open in general.