The Mathieu-moonshine twisted black-hole counting conjecture

Let $M_{24}$ be the Mathieu group, let $g\in M_{24}$, let $h\in C_{M_{24}}(g)$ be in the centralizer of $g$, and let $\Phi_{g,h}$ be the associated Siegel modular form. Define its reciprocal Fourier expansion by

$$\frac{1}{\Phi_{g,h}(\sigma,\tau,z)}=\sum_{m,n,\ell}D_{g,h}(m,n,\ell)p^m q^n y^\ell.$$

Let $\Omega_{1/4}^{g,h}(Q,P)$ be the index of $h$-twisted black-hole states in the $g$-orbifold CHL model.

Twisted black-hole counting conjecture. For every $g\in M_{24}$ and $h\in C_{M_{24}}(g)$,

$$\Omega_{1/4}^{g,h}(Q,P)=D_{g,h}\left(\frac{Q^2}{2},\frac{P^2}{2},Q\cdot P\right).$$

The conjecture connects generalized Mathieu moonshine functions with twisted BPS-state indices. The supplied status marks it disproved; the surrounding text says that the broader $M_{24}$ BPS-state symmetry question remained under investigation.