The facet conjecture for degree-zero differential-module Betti cones

Let $S=mathbbm{k}[x_1,\ldots,x_n]$ and $A=mathbbm{k}[t]$. Let $\operatorname{D^b_{DM}}(S)$ and $\operatorname{D^b_{DM}}(A)$ denote the relevant bounded derived categories of differential modules, let $\Phi(-,\mathcal{E})$ be the pairing with a vector bundle $\mathcal{E}$ on $mathbbm{P}^{n-1}$, and let $\tau_j$ and $\sigma_i$ be the linear functionals defining facets for $BS_{\text{DM}}(A,0)$. Facet conjecture. The exterior facets of the cone of Betti vectors of degree $0$ differential modules with finite-length homology over $S$ are given by the vanishing of linear functionals arising as the composition

$$\operatorname{D^b_{DM}}(S)\xrightarrow{\Phi(-,\mathcal{E})}\operatorname{D^b_{DM}}(A)\xrightarrow{f}\mathbbm{Z},$$

where $\mathcal{E}$ is a supernatural vector bundle on $mathbbm{P}^{n-1}$ and $f$ is one of the functionals $\tau_j$ or $\sigma_i$. Computations suggest this description of the facets and motivate a geometric extension of the one-variable facet theory, but the general assertion remains open.