The orbit-inequality conjecture for automorphic forms on Hodge-type Shimura varieties

Let $S_K$ be the special fiber of a Hodge-type Shimura variety at a prime $p$ of good reduction which splits in the reflex field $\mathbf E$. Let $G$ be the attached reductive group over $\mathbb F_p$, assumed split over $\mathbb F_{p^2}$, and let $\mathbf L$ be the Levi subgroup determined by the Hodge cocharacter. Write $W_{\mathbf L}$ for its Weyl group, and let $\Phi^+$ and $\Phi^+_{\mathbf L}$ denote the positive roots of $G$ and $\mathbf L$. For a weight $\lambda\in X^*(\mathbf T)$, a nonzero form $f\in H^0(S_K,\mathcal V_I(\lambda))$ has weight $\lambda$. The orbit-inequality conjecture. For every $W_{\mathbf L}\rtimes\operatorname{Gal}(\mathbb F_{p^2}/\mathbb F_p)$-orbit $\mathcal O\subset\Phi^+\setminus\Phi^+_{\mathbf L}$ and every subset $S\subset\mathcal O$, one has

$$\sum_{\alpha\in\mathcal O\setminus S}\langle\lambda,\alpha^\vee\rangle+\frac{1}{p}\sum_{\alpha\in S}\langle\lambda,\alpha^\vee\rangle\leq 0.$$

Equivalently, the space $H^0(S_K,\mathcal V_I(\lambda))$ vanishes outside the locus defined by these inequalities. This is an explicit consequence predicted by the zip-cone conjecture and gives an upper bound on possible weights; the paper presents it as conjectural because the underlying equality of saturated cones is not known in general.