Geometric automorphic forms conjecture for the zip cone

Let $C_{\operatorname{GS}}$ denote the cone of weights arising from the geometric automorphic forms considered in the source, and let $C_{\operatorname{zip}}$ be the cone of characters whose associated vector bundle on the stack of $G$-zips has a nonzero global section. Geometric zip-cone conjecture. One has

$$C_{\operatorname{GS}}\subset \langle C_{\operatorname{zip}}\rangle.$$

The claim is motivated by the expected closedness of the real saturated zip cone and by the inclusion of the geometric cone in the saturated cone of weights on the Shimura variety. Its general status is left open in the source.