Type-D0 constituent conjecture for Thompson moonshine modules

Let ${\rm Th}$ be the Thompson sporadic group, let ${\cal F}_3$ be the relevant weight one-half weakly holomorphic modular form, and let $W_m$ be the representation attached to the coefficient of $q^m$. Let $D_0(m)$ be a negative fundamental discriminant satisfying the two conditions of Proposition 5.7, and call an irreducible representation of ${\rm Th}$ of type $D_0$ when its character values generate the ring of algebraic integers in $\mathbb{Q}(\sqrt{D_0})$. Type-$D_0$ constituent conjecture. Whenever $-3m$ is a discriminant of ${\cal F}_3$ and

$$-3m=D_0(m)\lambda^2,$$

with $|D_0(m)|$ an order of ${\rm Th}$ and $(\lambda,3)=1$, the representation $W_m$ has at least one pair of irreducible representations of type $D_0$ as irreducible constituents. This is presented as the natural Thompson-group analogue of the cited dual-pair conjecture; the paper notes supporting evidence from the tabulated multiplicities, but does not establish it in general.