Singular-weight dimension conjecture for E8-invariant holomorphic Jacobi forms

Let $H(t)$ be the dimension of the space of $W(E_8)$-invariant holomorphic Jacobi forms of weight $4$ and positive index $t$. Let $N(t)$ be the number of distinct Weyl orbits of vectors of norm $t$, where norm means $\frac{1}{2}(v,v)=t$. For a Weyl orbit $\operatorname{orb}(m)$ of norm $t$, write $\operatorname{orb}(m)$ for the corresponding orbit sum in a Fourier expansion. Singular-weight conjecture.

$$H(t)=N(t).$$

Equivalently, for every such Weyl orbit there exists a unique $W(E_8)$-invariant holomorphic Jacobi form of weight $4$ and index $t$ with Fourier expansion

$$\Phi_{t,m}=1+\frac{240}{|\operatorname{orb}(m)|}q\cdot\operatorname{orb}(m)+O(q^2).$$

The conjecture has been proved for indices $t\leq11$; the source reports consistency with computations at indices $12$ and $13$, but the general statement remains open.