Rational toral rank conjecture

Let $X$ be a nilpotent finite CW-complex. Its rational torus rank ${\operatorname{rk}}_0(X)$ is the largest integer $n$ for which a torus $T^n$ acts almost freely on a finite CW-complex having the same rational homotopy type as $X$. Rational toral rank conjecture. The total rational cohomology dimension satisfies

$$\dim_{\mathbb{Q}} H^*(X,\mathbb{Q})\geq 2^{{\operatorname{rk}}_0(X)}.$$

This is the rational formulation of the Halperin–Carlsson conjecture and connects torus actions with Sullivan models and rational homotopy theory. The source presents it as a longstanding conjecture.