Sturmfels's conjecture on exceptional parameters for A-hypergeometric systems

Let $A$ be a homogeneous matrix, let $I_A$ be its underlying toric ideal, and let $H_A(\beta)$ be the $A$-hypergeometric system with parameter vector $\beta\in\mathbb{C}^d$. Its holonomic rank is the dimension of the local holomorphic solution space at a nonsingular point. Define the exceptional set by

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

Sturmfels's conjecture. The exceptional set is empty if and only if $I_A$ is Cohen–Macaulay. For Cohen–Macaulay $I_A$, the rank equals $\operatorname{vol}(A)$ for every parameter, whereas in general it is at least $\operatorname{vol}(A)$ and equals this value for generic parameters. The conjecture identifies the absence of exceptional parameters with the Cohen–Macaulay property of the toric ideal.