The exceptional-set characterization of Cohen–Macaulay toric ideals

Let $A$ be a matrix defining a toric ideal $I_A$ and an $A$-hypergeometric system $H_A(\beta)$. Its exceptional set is

$$\mathcal{E}(A)=\{\beta\in\mathbb{C}^d:\operatorname{rank}(H_A(\beta))>\operatorname{vol}(A)\}.$$

Here $\operatorname{vol}(A)$ is the normalized volume of the configuration $A$.

Exceptional-set characterization. The exceptional set $\mathcal{E}(A)$ is empty if and only if the toric ideal $I_A$ is Cohen–Macaulay.

In dimension two, the exceptional set is known explicitly and this equivalence follows from the result of Cattani, D’Andrea, and Dickenstein. The conjecture proposes that the same characterization holds for arbitrary matrices $A$; the supplied text gives experimental evidence but does not state a resolution.