Free actions on products of real projective spaces

We prove that if $G=(\mathbb{Z}\mathrm{/}2{)}^r$ acts freely and cellularly on a finite-dimensional CW-complex $X$ homotopy equivalent to $\mathbb{R}{P}^{n_1}\times \cdots \times \mathbb{R}{P}^{n_k}$ with trivial action on the mod-$2$ cohomology, then $r\le \mu (n_1)+\cdots +\mu (n_k)$ where for each integer $n\ge 0$, $\mu (n)=0$ if $n$ is even, $\mu (n)=1$ if $n\equiv 1$ mod 4, and $\mu (n)=2$ if $n\equiv 3$ mod 4. This proves a homotopy-theoretic version of a conjecture of Cusick.