Odd-multiplicity type-D0 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 $D_0(m)\in\{-15,-24,-39\}$ and let $V\oplus\overline V$ denote a dual pair of irreducible representations of type $D_0$. Odd-multiplicity type-$D_0$ conjecture. If $-3m$ is a discriminant of ${\cal F}_3$ and

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

with $(\lambda,3)=1$, then $W_m$ contains a pair $V\oplus\overline V$ of irreducible representations of type $D_0$ with odd multiplicity. This is proposed as an analogue of the cited conjecture concerning odd-multiplicity decompositions; no resolution is given in the paper.