Halperin–Carlsson rank conjecture for semifree differential graded modules

Let $k$ be an algebraically closed field of characteristic $2$, let $S=k[x_1,\ldots,x_r]$ be the polynomial algebra in $r$ variables of degree $-1$, and let $(M,\partial)$ be a differential graded $S$-module if $M$ is an $S$-module and $\partial$ is an $S$-linear endomorphism of degree $-1$ satisfying $\partial^2=0$. Suppose that $(M,\partial)$ is semifree, meaning that its underlying graded $S$-module is free. Halperin–Carlsson rank conjecture. If the homology of $(M,\partial)$ is nonzero and finite dimensional as a $k$-vector space, then

$$\operatorname{rank}_S M\geq 2^r.$$

This algebraic rank bound is a stronger version of the Halperin–Carlsson conjecture for free $2$-torus actions. The paper discusses multiplicative structures on minimal Hirsch–Brown models and recovers or generalizes results of Carlsson and Allday–Puppe; the supplied text does not establish whether this conjecture has been resolved.