The discriminant dual-pair conjecture for umbral modules

Let $H^{(\ell)}$ be an umbral vector-valued mock modular form, let $D$ be a discriminant of $H^{(\ell)}$, and let $K^{(\ell)}_{r,-D/4\ell}$ be the corresponding representation of $G^{(\ell)}$. Suppose that $D=-n\lambda^2$ for some integer $\lambda$, where $n$ is of the type considered in the source. A dual pair means a pair of dual irreducible representations, both of type $n$, occurring as irreducible constituents. Discriminant dual-pair conjecture. The representation $K^{(\ell)}_{r,-D/4\ell}$ contains at least one dual pair of irreducible representations of type $n$. The conjecture extends the preceding proposition from the smallest relevant discriminants to all discriminants of this form; it remains open in the source.