Finite-generation conjecture for Weyl-invariant E8 weak Jacobi forms

Let $W(E_8)$ be the Weyl group of $E_8$, and consider all $W(E_8)$-invariant weak Jacobi forms of integral weight and integral index. Let $M_*(\operatorname{SL}_2(\mathbb Z))$ denote the graded algebra of modular forms for $\operatorname{SL}_2(\mathbb Z)$. Finite-generation conjecture. The algebra of all $W(E_8)$-invariant weak Jacobi forms of integral weight and integral index is finitely generated over $M_*(\operatorname{SL}_2(\mathbb Z))$. Finite generation is known for weak Jacobi forms attached to arbitrary rank-two lattices, which motivates this conjecture; the E8 statement itself is presented as unresolved.