The type-$n$ dual-pair conjecture for umbral modules

Let $X$ be a Niemeier root system with Coxeter number $m$, and let $D$ be a discriminant of $H^X$. Dual-pair conjecture. If

$$D=-n\lambda^2$$

for some integer $\lambda$, then the representation $K^X_{r,-D/4m}$ has at least one dual pair of irreducible representations of type $n$ among its irreducible constituents. The conjecture extends the observed smallest-discriminant phenomenon; it was verified in the source for $X=A_1^{24}$, while the general statement remains open.