Goldring–Koskivirta's cone conjecture for Hodge type Shimura varieties

Let $\overline{\operatorname{Sh}}^{\operatorname{tor}}$ be the toroidal compactification of the Shimura variety, let $T$ be the relevant maximal torus, and let $X^*(T)$ be its character group. For $\lambda\in X^*(T)$, write $\omega(\lambda)$ for the corresponding automorphic vector bundle, and let $\underline{W(\lambda)}$ be the vector bundle on the stack $G\text{-Zip}$ whose pullback to $\operatorname{Sh}$ is $\omega(\lambda)$. Define

$$C_{\overline{\operatorname{Sh}}}=\{\lambda\in X^*(T):\text{there exists }n\geq 1\text{ such that }H^0(\overline{\operatorname{Sh}}^{\operatorname{tor}},\omega(n\lambda))\neq 0\},$$

and

$$C_{\operatorname{Zip}}=\{\lambda\in X^*(T):\text{there exists }n\geq 1\text{ such that }H^0(G\text{-Zip},\underline{W(n\lambda)})\neq 0\}.$$

Goldring–Koskivirta's cone conjecture. The two cones coincide:

$$C_{\overline{\operatorname{Sh}}}=C_{\operatorname{Zip}}.$$

This expected equality relates the weights of automorphic forms extending to the toroidal compactification to the weights producing sections on the associated $G$-Zip stack. The paper presents it as a general expected theorem used as an assumption because the required vanishing and coherent-cohomology results are not available in full generality.