Free-rank conjecture for products of spheres

Let $p$ be a prime, let $n_1,\ldots,n_k$ be positive integers, and let $(\mathbb{Z}_p)^r$ act freely on the product $S^{n_1}\times\cdots\times S^{n_k}$. Free-rank conjecture for products of spheres. One should have

$$r\leq k.$$

This is a generalization of the spherical space-form problem. The statement is known in several special cases, including homologically trivial actions on products of equidimensional spheres and certain high-dimensional regimes, but the general case remains open.