Carlsson's toral rank conjecture

Let $p$ be a prime, let $G=({\Bbb Z}/p)^k$, and let $X$ be a finite free $G$-complex. Carlsson's toral rank conjecture. One has

$$\sum_i\dim_{{\Bbb Z}/p}H_i(X,{\Bbb Z}/p)\geq 2^k.$$

This conjecture gives a lower bound on the total mod-$p$ homology of a finite free complex in terms of the rank of the elementary abelian group acting freely. The supplied text attributes it to G. Carlson and does not provide evidence of a resolution, so its status is recorded as open.