Finite generation conjecture for the ring of automorphic forms on the stack of $G$-zip flags

Let $G$ be a reductive group with zip datum determined by $\mu$, let $T$ be a maximal torus, and for each character $\lambda\in X^*(T)$ let ${\mathcal V}_I(\lambda)$ be the associated vector bundle on \mathop{\text{G-\tt Zip}}\nolimits^\mu. Define the graded $k$-algebra

$$R_{\mathsf{zip}}=\bigoplus_{\lambda\in X^*(T)}H^0(\mathop{\text{$G$-\tt Zip}}\nolimits^\mu,{\mathcal V}_I(\lambda)).$$

Finite generation conjecture. The ring $R_{\mathsf{zip}}$ is a finitely generated $k$-algebra; equivalently, the stack \mathop{\text{G-\tt ZipFlag}}\nolimits^\mu is a Mori dream space.

The conjecture concerns finite generation of the ring of higher-rank mod $p$ automorphic forms. The paper verifies a few cases by hand, while no general resolution is stated.