The stable Carlsson conjecture for manifolds

Let $M$ be a closed connected manifold, and let $\operatorname{rank}_p(M)$ denote the largest integer $r$ such that $(\mathbb{Z}/p)^r$ acts freely on $M$.

Stable Carlsson conjecture. There exists a constant $C$ such that, for every prime $p>C$,

$$\dim H^*(M,\mathbb{Z}/p)\geq 2^{\operatorname{rank}_p(M)}.$$

This is a weaker, stable version of Carlsson's conjecture: the asserted bound is required only for primes larger than a constant depending on $M$. The source does not specify whether it is open or resolved.