The doublet and sextet conjecture for umbral modules

Let $X$ be a Niemeier root system with Coxeter number $m$. A representation is a doublet if it is the direct sum of two copies of one representation, and a sextet if it is the direct sum of six copies of one representation. Doublet and sextet conjecture. The representation $K^X_{r,-D/4m}$ is a doublet if and only if $D\ne0$ and $D\ne-n\lambda^2$ for every integer $\lambda$ and every $n$ listed in the source's discriminant table for $X$. If $X=A_8^3$, then $K^X_{r,-D/4m}$ is a sextet if and only if

$$D\ne-27\lambda^2$$

for some integer $\lambda$. The source presents this as a conjecture based on observed multiplicity patterns; no resolution is supplied.