Upper semicontinuity conjecture for Gevrey slopes of $A$-hypergeometric systems

Let $A$ be the defining matrix, let $s>1$ be a Gevrey order, and let ${\operatorname{d}_s}(A,\beta)$ denote the corresponding Gevrey irregularity invariant for parameter $\beta\in\mathbb C^d$. Write $S_A^{\{n\}}$ for the semigroup module associated with the face $\{n\}$, and set

$$\varepsilon_A=\sum_{i=1}^n a_i.$$

Gevrey-slope conjecture. The map $\beta\mapsto {\operatorname{d}_s}(A,\beta)$ is upper-semicontinuous. Moreover,

$${\mathcal E}_A^{n}(s)=\{\beta\in\mathbb C^d\mid {\operatorname{d}_s}(A,\beta)>{\operatorname{d}_s}(A)\}=-\operatorname{qdeg}\left(\bigoplus_{q=0}^{d-1}\operatorname{Ext}^{n-q}_{\mathbb C[\partial]}(S_A^{\{n\}},\mathbb C[\partial])(-\varepsilon_A)\right),$$

and, in particular, ${\mathcal E}_A^{n}(s)=\varnothing$ if and only if $S_A^{\{n\}}$ is Cohen--Macaulay. The preceding theorem proves upper semicontinuity under an additional hypothesis on the arrangement of the relevant hyperplanes, while the general assertion and the stated description of the exceptional set are left as conjectural.