The umbral module doublet conjecture

Let $\Lambda=\{2,3,4,5,7,13\}$ and let $K^{(\ell)}_{r,-D/4\ell}$ be the corresponding representation of $G^{(\ell)}$. Say that a $G$-module is a doublet if it is isomorphic to the direct sum of two copies of a single representation. Umbral module doublet conjecture. The representation $K^{(\ell)}_{r,-D/4\ell}$ is a doublet if and only if

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

for every integer $\lambda$ and every $n$ satisfying the conditions of the cited proposition. This conjecture predicts precisely when the conjectural graded pieces are doublets; it remains open in the source.