The virtual projective dimension bound for smooth toric varieties

Let $X=X(\Sigma)$ be a smooth toric variety with irrelevant ideal $B$, let $S$ be its Cox ring, and let $I$ be a $B$-saturated ideal. Define the virtual projective dimension of an $S$-module $M$ by

$$\operatorname{vpdim} M=\min\left\{\operatorname{length}(F_{\bullet})\mid F_{\bullet}\text{ a virtual resolution of }M\right\}.$$

Virtual projective dimension bound. One should have

$$\operatorname{vpdim} S/I\leq n.$$

This asks whether the analogue of the Hilbert syzygy bound for products of projective spaces extends to arbitrary smooth toric varieties. The supplied text does not state whether the conjecture has been resolved.