The top-dimensional cohomology surjectivity conjecture for elementary abelian actions

Let $G=(\mathbb Z/2)^r$ act freely on

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

and let $N=n_1+\dots+n_m$. Write $\pi_G:X/G\to BG$ for the classifying map of the action. Top-dimensional surjectivity conjecture. The induced map

$$\pi_G^*:H^N(G,\mathbb F_2)\to H^N(X/G,\mathbb F_2)$$

is surjective. This conjecture would imply the rank conjecture for products of spheres in the elementary abelian $2$-group case. The source gives examples where the required surjectivity is known, but does not report a general resolution.