Van den Bergh's NCCR existence conjecture for affine Gorenstein toric varieties

Let $X_\sigma=\operatorname{Spec} R_\sigma$ be an affine Gorenstein toric variety, where $R_\sigma=k[\sigma^\vee\cap M]$ is the toric algebra associated to a full-dimensional cone $\sigma\subset N_\mathbb{R}$, and suppose that there is an element $m\in M$ satisfying $\langle m,u_{\rho_i}\rangle=1$ for every primitive generator $u_{\rho_i}$ of a ray of $\sigma$. An NCCR is a non-commutative crepant resolution, namely an algebra of the form $\Lambda=\operatorname{End}_{R_\sigma}(M')$ for a finitely generated reflexive $R_\sigma$-module $M'$ such that $\Lambda$ has finite global dimension and is maximal Cohen–Macaulay as an $R_\sigma$-module.

Van den Bergh's NCCR existence conjecture. Every affine Gorenstein toric variety has an NCCR.

The conjecture concerns the existence of non-commutative counterparts to crepant resolutions. The source states that it remains open, although NCCRs are known in a number of cases.