Weak Horrocks conjecture for finite-dimensional modules

Let $S(m)=\mathbb Q[v_1,\dots,v_m]$ be a polynomial ring, and let $M$ be a finite-dimensional graded $S(m)$-module. Here $\operatorname{Tor}_{S(m)}^*(M,\mathbb Q)$ denotes the total Tor space, and $\mathop{\mathrm{dim}}$ is dimension over $\mathbb Q$.

Weak Horrocks conjecture. One has

$$\mathop{\mathrm{dim}} \operatorname{Tor}_{S(m)}^*(M,\mathbb Q)\geqslant 2^m.$$

This is presented as a weaker variant of the preceding Horrocks conjecture and is relevant to estimating the cohomology rank of spaces with torus actions. The source does not provide evidence of a resolution for this weak variant.