The cone conjecture for automorphic forms on schemes over a G-zip

Let $S$ be a $k$-scheme with a morphism

$$\zeta\colon S\to \mathop{\text{$G$-\tt Zip}}\nolimits^\mu.$$

Assume that $\zeta$ is smooth, its restriction to every connected component of $S$ is surjective, and, for every $w\in W$ with $\ell(w)=1$, the closure $\overline{\operatorname{Flag}(S)}_w$ is pseudo-complete, meaning that every global regular function is Zariski locally constant. Define

$$C_S=\{\lambda\in X^*(T)\mid H^0(S,\mathcal V_I(\lambda))\neq 0\}$$

and let $C_{\operatorname{zip}}$ be the corresponding cone of weights with nonzero global sections on \mathop{\text{G-\tt Zip}}\nolimits^\mu. For a cone $C\subset X^*(T)$, write

$$\langle C\rangle=\{\lambda\in X^*(T)\mid \exists N\geq 1,\ N\lambda\in C\}.$$

The cone conjecture. Under these assumptions,

$$\langle C_S\rangle=\langle C_{\operatorname{zip}}\rangle.$$

The result generalizes the Shimura-variety formulation and would identify the saturated cone of automorphic-form weights on $S$ with the intrinsic zip cone. The source presents it as a conjectural statement under the stated geometric assumptions.