Avramov–Buchweitz–Iyengar rank conjecture for differential modules

Let $R$ be a regular local ring of dimension $d$, and let $F$ be a differential $R$-module admitting a finite free flag. If the homology $H(F)=\ker\delta/\operatorname{im}\delta$ has non-zero finite length, then Avramov–Buchweitz–Iyengar's rank conjecture.

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

This conjecture extends the lower-bound problem for finite free complexes to differential modules with finite free flags. The paper is motivated by this conjecture and establishes the corresponding lower bound for multigraded differential modules over polynomial rings, while the stated local-ring conjecture remains unresolved here.