The rank conjecture for differential modules with finite free flag

Let $R$ be a local ring and let $F$ be a differential $R$-module admitting a finite free flag. Suppose that $\operatorname{H}(F)$ has non-zero finite length, and write $d=\dim R$. The rank conjecture. One has

$$\operatorname{rank}_{R}F\geq 2^d.$$

The conjecture is motivated by rank bounds for finite free resolutions and by Carlsson's conjecture in transformation-group topology. The surrounding text records several partial results, but does not resolve the assertion in general.