Sun and Wang's lower-bound conjecture for weights of $W(E_8)$-invariant weak Jacobi forms

Let $m$ be an index and let $k$ be the weight of a $W(E_8)$-invariant weak Jacobi form. Sun and Wang's lower-bound conjecture. Every non-zero $W(E_8)$-invariant weak Jacobi form of index $m$ has weight at least $-4m$; equivalently, its weight satisfies $k\geq -4m$.

The conjecture is supported in the paper by constructing all such forms with weight $k\leq -4m$ and index $m$ for $m\leq 28$, and finding none with $k<-4m$ in that range. Its general status is not resolved by the supplied evidence.