The flattening surjectivity conjecture for differential-module Betti cones

Let $S=mathbbm{k}[x_1,\ldots,x_n]$ be a polynomial ring over a field $mathbbm{k}$. For an integer $a$, let $BS_{\text{mod}}(S)$ be the rational cone of Betti tables of graded finite-length $S$-modules, and let $BS_{\text{DM}}(S,a)$ be the rational cone of Betti vectors of degree $a$ differential $S$-modules with finite-length homology. Regard a minimal free resolution as a differential module via the folding functor, which induces a flattening map from Betti tables to Betti vectors. Flattening surjectivity conjecture. This flattening gives a surjection of cones

$$BS_{\text{mod}}(S)\to BS_{\text{DM}}(S,a)$$

for every $a$. More precisely, every differential module with finite-length homology has a Betti vector that is a positive rational combination of Betti vectors of differential modules whose homology has Betti tables extremal in $BS_{\text{mod}}(S)$. The conjecture extends the Boij–Söderberg theory from graded modules and resolutions to differential modules; the paper presents examples and partial results as evidence, while the full assertion remains open.