The saturated zip-cone conjecture for Hodge-type Shimura varieties

Let $S_K$ be the special fiber of a Hodge-type Shimura variety at a prime $p$ of good reduction, and let $C_K(\overline{\mathbb{F}}_p)$ be the cone of weights of nonzero automorphic forms on $S_K$. Let $(G,\mu)$ be the associated cocharacter datum and let $C_{\operatorname{zip}}$ be its zip cone. For a cone $C$, write

$$\langle C\rangle=\{\lambda\mid \exists N\geq 1,\ N\lambda\in C\}.$$

The saturated zip-cone conjecture. One has

$$\langle C_K(\overline{\mathbb{F}}_p)\rangle=\langle C_{\operatorname{zip}}\rangle.$$

This predicts that the saturated cone of automorphic weights on a Shimura variety is determined by the associated $G$-zip. The paper presents this as a conjecture motivated by the theory of automorphic forms in characteristic $p$; no resolution is supplied here.