Stability conjecture for E8 weak Jacobi-form generator counts

For each index $t$, let $d_{k,t}$ be the number of weight-$k$ generators of the free module $J_{*,E_8,t}^{\operatorname{w},W(E_8)}$ of $W(E_8)$-invariant weak Jacobi forms. Stability conjecture. For every even integer $K\leq0$, there exists a positive integer $L(K)$ such that, for each fixed $k$ with $K\leq k\leq0$, the number $d_{k,t}$ is constant for all $t\geq L(K)$. This predicts eventual stability of the generator multiplicities in every fixed non-positive weight range; the source gives no resolution.