Goldring–Koskivirta conjecture on global sections over G-zips

Let $G$ be a reductive group over a finite field $\mathbb{F}_q$, let $k=\overline{\mathbb{F}}_q$, and let $\mu\colon\mathbb{G}_{\mathrm{m},k}\to G_k$ be a cocharacter. Let \mathop{\text{G-\tt Zip}}\nolimits^\mu be the stack of $G$-zips of type $\mu$, and let ${\mathcal C}_{\mathsf{zip}}$ be the saturation of the cone of $\lambda\in X^*(T)$ for which ${\mathcal V}_I(\lambda)$ has a nonzero section on this stack. For a Hodge-type Shimura variety with special fiber $S_K$, let ${\mathcal C}(\overline{\mathbb{F}}_p)$ be the saturation of the cone of weights with nonzero global sections on $S_K$. Goldring–Koskivirta conjecture. One has

$${\mathcal C}(\overline{\mathbb{F}}_p)={\mathcal C}_{\mathsf{zip}}.$$

For Hilbert–Blumenthal Shimura varieties this equality is known, with the common cone generated by the weights of partial Hasse invariants. In general, the equality is presented as a conjectural description of the saturated cone of automorphic forms via the geometry and representation theory of $G$-zips.