Goldring–Koskivirta conjecture for flag spaces over G-zips

Let $k$ be a field, let $G$ be a reductive group over a finite field, let $\mu\colon\mathbb{G}_{\mathrm{m},k}\to G_k$ be a cocharacter, and let $(X,\zeta)$ consist of a $k$-scheme $X$ and a smooth, surjective morphism \zeta\colon X\to\mathop{\text{G-\tt Zip}}\nolimits^\mu. Let $(Y,\zeta_{\operatorname{flag}})$ be the associated flag space. Assume that for every $w\in W$ with $\ell(w)=1$, the closed stratum $\overline{Y}_w$ is pseudo-complete, and that the restriction of $\zeta$ to every connected component $X^\circ\subset X$ is smooth and surjective. Let ${\mathcal C}_Y$ be the saturated cone of characters whose associated vector bundles on $Y$ have nonzero global sections, and let ${\mathcal C}_{\mathsf{zip}}$ be the saturated cone attached to the stack of $G$-zips. Goldring–Koskivirta flag-space conjecture. Under these assumptions, one has

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

The conjecture would identify global sections on the flag space with the group-theoretical cone from $G$-zips. The source notes that the inclusion from partial Hasse-invariant weights to the zip cone can be strict in general; the conjecture is intended to characterize when the flag space has the Hasse property.