Minimal-weight conjecture for Weyl-invariant E8 weak Jacobi forms

Let $t$ be an index, and let $W(E_8)$ denote the Weyl group of the $E_8$ root system. A weak Jacobi form is allowed to have Fourier terms with negative discriminant, and its weight is the modular weight. Minimal-weight conjecture. Every non-zero $W(E_8)$-invariant weak Jacobi form of index $t$ has weight at least $-4t$. The conjecture sharpens the known lower bound of $-5t$ obtained by pull-back to $W(E_7)$-invariant Jacobi forms; the source does not state a resolution.