The categorical duality conjecture for degree-zero differential-module Betti vectors

Let $S=mathbbm{k}[x_1,\ldots,x_n]$ and $A=mathbbm{k}[t]$. Let $mathbbm{B}$ be the vector space of Betti vectors, and let $\mathcal{E}$ be a vector bundle on $mathbbm{P}^{n-1}$ with absolute Hilbert function. Categorical duality conjecture. For $\mathbf{b}inmathbbm{B}$, the following are equivalent: (1) $\mathbf{b}=\beta^{\text{DM}}(F)$ for a degree $0$ differential module $F$ over $S$ with finite-length homology; (2) for every vector bundle $\mathcal{E}$ on $mathbbm{P}^{n-1}$, $\mathbf{b}$ pairs with the absolute Hilbert function of $\mathcal{E}$ to give the Betti vector of a differential module in $\operatorname{DM}(A,0)$ with finite-length homology. This conjecture seeks a differential-module analogue of the Boij–Söderberg duality between Betti and cohomology cones; the paper constructs the relevant pairing, but the asserted equivalence is open.