Finite generation conjecture for the automorphic-form algebra on the stack of G-zips

Let $R_{\rm zip}$ be the graded algebra

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

where $\mathscr V(\lambda)$ is the automorphic vector bundle associated with $\lambda$. Finite generation conjecture. The $\overline{\mathbf F}_p$-algebra $R_{\rm zip}$ is finitely generated. This is a stronger algebraic statement than finite generation of the associated weight cone. The source presents it as a conjecture, and no resolution is supplied.