The umbral moonshine Rademacher-sum conjecture

Let $\Lambda=\{2,3,4,5,7,13\}$, let $G^{(\ell)}$ be the corresponding umbral group, and let $H^{(\ell)}_g$ be its vector-valued mock modular McKay–Thompson series. For $g\in G^{(\ell)}$, let $n_g$ and $h_g$ be the associated positive integers, and let $R^{(\ell)}_{n|h}$ denote the regularized vector-valued Rademacher sum for the weight-$1/2$ action on $(\ell-1)$-component functions. Umbral moonshine conjecture. For every $\ell\in\Lambda$ and $g\in G^{(\ell)}$,

$$H^{(\ell)}_g=R^{(\ell)}_{n|h},\qquad n=n_g,\quad h=h_g,$$

where

$$R^{(\ell)}_{n|h}=\operatorname{Reg}\left(\sum_{\gamma\in\Gamma_{\infty}\backslash\Gamma_0(n)}\left.\begin{pmatrix}-2q^{-1/4\ell}\\0\\vdots\\0\end{pmatrix}\right|_{1/2,n|h}\gamma\right).$$

The conjecture extends the Mathieu-group Rademacher-sum description to all umbral groups; its general validity is not established in the source.