Free elementary abelian action conjecture for products of odd-dimensional spheres

Let $p$ be a prime, let $k\geq 0$ be an integer, and let $n_1,\ldots,n_k$ be integers. For a space $X$, define its free $p$-rank by

$$\operatorname{frk}_p(X)=\max\{r\mid (\mathbb{Z}/p)^r\text{ acts freely on }X\}.$$

Free-rank conjecture. One has

$$\operatorname{frk}_p(\mathbb{S}^{2n_1+1}\times\cdots\times\mathbb{S}^{2n_k+1})=k$$

for each prime $p$ and integer $k\geq 0$. This is presented as a consequence of the Benson–Carlson conjecture and as a conjecture concerning free actions of elementary abelian $p$-groups on products of odd-dimensional spheres. The source records several partial results but does not state a complete resolution.