The umbral module factoring-through conjecture

Let $X$ be a Niemeier root system, with associated groups $G^X$ and $\bar G^X$, and let $K^X_r$ denote the $\mathbb{Q}$-graded component indexed by $r\in\mathbb{Z}/2m\mathbb{Z}$. Set

$$c=\frac{\#G^X}{\#\bar G^X}.$$

Factoring-through conjecture. The $\mathbb{Q}$-graded $G^X$-module $K^X_r$ is a faithful representation of $G^X$ when $r\equiv0\pmod c$, and factors through $\bar G^X$ otherwise. This is presented as a consequence of the umbral module conjecture and the paired McKay--Thompson relations; its general validity remains open.