The umbral McKay–Thompson shadow conjecture

Let $X$ be a Niemeier root system with Coxeter number $m$, let $G^X$ be its associated group, and let $S^X_g$ be the attached cusp form for $g\in G^X$. For $g\in G^X$, let $n_g$ be the order of its image in $\bar G^X$, and let $h_g=N_g/n_g$ as defined from the associated permutation cycle lengths. Shadow conjecture. For fixed $X$ and $g$, the graded super-characters of the components indexed by $r\in\mathbb{Z}/2m\mathbb{Z}$ define the components of a vector-valued mock modular form $H^X_g$ of weight $1/2$ on $\Gamma_0(n_g)$ with shadow $S^X_g$. This conjecture identifies the cusp forms attached to the group action as the shadows of the conjectural umbral McKay--Thompson series; the general statement remains open.