The cone conjecture for global sections on G-zip period spaces

Let $k$ be a field, let $X$ be a proper $k$-scheme, and let $\zeta\colon X\to G\text{-}{\tt Zip}^\mu$ be smooth and surjective on each connected component. For $\lambda\in X^*(T)$, let ${\mathcal V}_I(\lambda)=\zeta^*({\mathcal V}_I(\lambda))$ and define

$$C_X=\left\{\lambda\in X^*(T)\mid H^0(X,{\mathcal V}_I(\lambda))\neq 0\right\}.$$

Let ${\mathcal C}_X$ be the saturation of $C_X$, and let ${\mathcal C}_{\mathsf{zip}}$ denote the corresponding zip cone.

Cone conjecture. Under these assumptions, one has

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

This is formulated for period maps arising, for example, from smooth toroidal compactifications of Hodge-type Shimura varieties. The statement predicts that the saturated cone of weights with nonzero global sections is determined universally by the stack of $G$-zips.