The toral rank and Carlsson conjectures for manifolds

Let $M$ be a closed connected manifold. Its toral rank is the largest integer $r$ such that $T^r$ acts almost freely on $M$, and, for a prime $p$, its $p$-rank is the largest integer $r$ such that $(\mathbb{Z}/p)^r$ acts freely on $M$.

Toral rank and Carlsson conjectures.

$$\dim H^*(M,\mathbb{Q})\geq 2^{\operatorname{rank}(M)}$$

and, for every prime $p$,

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

These conjectures, attributed respectively to S. Halperin and G. Carlsson, relate the ranks of torus and elementary abelian group actions to the total cohomology of the manifold. Their resolution status is not specified in the source.