The Mathieu moonshine conjecture for twisted elliptic genera

For each $g\in M_{24}$, let ${\cal Z}_g(\tau,z)$ have Fourier expansion

$${\cal Z}_g(\tau,z)=\sum_{n\geq0,\,\ell\in\mathbb{Z}}c_g(4n-\ell^2)q^ny^\ell.$$

Let $\hat K=\left(\bigoplus_{\ell=0}^{\infty}\hat K_{4\ell-1}\right)\bigoplus\left(\bigoplus_{\ell=0}^{\infty}\hat K_{4\ell}\right)$ be an infinite-dimensional, $\mathbb{Z}$-graded super (or virtual) $M_{24}$-module, and let $\operatorname{str}_{\hat K_k}g$ denote the supertrace.

Twisted elliptic-genus module conjecture. For $k\equiv0,3\pmod4$ and $k\geq-1$, one has

$$c_g(k)=\operatorname{str}_{\hat K_k}g.$$

For $k\neq0$, $\hat K_k$ is even, of the form $\hat k_k\oplus\hat k_k^*$ for a super representation $\hat k_k$ and its dual. This conjecture refines the Mathieu moonshine module picture by encoding the twisted $K3$ elliptic-genus coefficients as supercharacters; the displayed coefficients and corresponding modules are computed in the source, while the general module interpretation is the conjectural part.