The virtual-resolution minimality conjecture for multigraded regular modules

Let $S$ be the Cox ring of a product of projective spaces, let $B$ be its irrelevant ideal, and let $M$ be a multigraded $S$-module. For a multidegree $\mathbf d$, write $M(\mathbf d)$ for the corresponding shift, $M_{\geq\mathbf d}$ for its multigraded truncation, and let $\mathcal{H}_B^i(M)$ denote the $i$th local cohomology module with support in $B$. Assume that $M$ is $\mathbf d$-regular and that

$$\mathcal{H}_B^0(M)=\mathcal{H}_B^1(M)=0.$$

Virtual-resolution minimality conjecture. The virtual resolution of $M(\mathbf d)$ from Proposition~ is the minimal free resolution of $M_{\geq\mathbf d}(\mathbf d)$.

The conjecture proposes that the linear-presentation hypothesis in Corollary~ is unnecessary. It would identify the virtual resolution obtained from multigraded regularity with the ordinary minimal free resolution of the truncated module under the stated local-cohomological vanishing conditions.