Goldring–Koskivirta Cone Conjecture for Hodge-type Shimura varieties

Let $S_K$ be a Hodge-type Shimura variety, let ${\mathcal C}_K$ be the saturation of the set of characters whose associated automorphic vector bundles on $S_K$ have nonzero degree-zero cohomology, and let ${\mathcal C}_{\mathsf{zip}}$ be the corresponding saturation defined using vector bundles on the stack of $G$-zips. Goldring–Koskivirta's Cone Conjecture. For any Hodge-type Shimura variety, one has

$${\mathcal C}_K={\mathcal C}_{\mathsf{zip}}.$$

In other words, the vanishing of degree-zero cohomology of automorphic vector bundles is encoded by the stack of $G$-zips. The conjecture was proposed by Goldring and Koskivirta; for Hilbert–Blumenthal Shimura varieties it is verified by Goldring and Koskivirta and proved independently by Diamond and Kassaei, while the statement for arbitrary Hodge-type Shimura varieties is the formulation appearing here.