The umbral moonshine shadow conjecture

Let $\Lambda=\{2,3,4,5,7,13\}$, let $g\in G^{(\ell)}$, and let $H^{(\ell)}_g=(H^{(\ell)}_{g,r})$ be the vector-valued mock modular form obtained from the graded supertrace functions for $0<r<\ell$. Let $S^{(\ell)}=(S^{(\ell)}_r)$ be the associated vector-valued theta series, let $\chi^{(\ell)}_{g,r}$ be determined by the twisted Euler characters, and let $n_g$ be the order of the image of $g$ in $\bar G^{(\ell)}$. Umbral moonshine shadow conjecture. For fixed $\ell\in\Lambda$ and $g\in G^{(\ell)}$, the functions $H^{(\ell)}_{g,r}$ are the components of a weight-$1/2$ vector-valued mock modular form on $\Gamma_0(n_g)$ with shadow

$$S^{(\ell)}_g=(S^{(\ell)}_{g,r})=(\chi^{(\ell)}_{g,r}S^{(\ell)}_r),$$

where $\chi^{(\ell)}_{g,r}=\bar\chi^{(\ell)}_g$ for odd $r$ and $\chi^{(\ell)}_{g,r}=\chi^{(\ell)}_g$ for even $r$. This conjecture specifies the modular shadows predicted for the graded supertraces; the source presents it as an unproved consequence of the broader umbral moonshine picture.