Berkesch–Erman–Smith's virtual resolution length conjecture

Let $Y$ be a smooth toric variety with Cox ring $S$, and let $M$ be a finitely generated graded $S$-module. A virtual resolution of $M$ is a complex of graded free $S$-modules whose associated complex of sheaves is a locally free resolution of the associated sheaf of $M$.

Berkesch–Erman–Smith's conjecture. The module $M$ admits a virtual resolution of length at most

$$\dim(Y).$$

This conjecture is a version of Hilbert's Syzygy Theorem for virtual resolutions. It was proved for products of projective spaces by Berkesch, Erman, and Smith, for monomial ideals in Cox rings of smooth toric varieties by Yang, and in the paper's setting for smooth projective toric varieties of Picard rank $2$.