The central-extension relabeling conjecture for Mathieu moonshine

Let $g\in M_{23}\subset M_{24}$, let $\alpha$ satisfy the cocycle conditions referred to in the source, and let $C^\alpha_{M_{24}}(g)$ be the corresponding central extension, with elements $Q$ and $g$ as in the construction. For the twisted twining genera $\phi_{a,b}(\tau,z)$, and for $k\in C^\alpha_{M_{24}}(g)$ and $m,r\in\mathbb Z$, central-extension relabeling conjecture. There is an automorphism $\varphi:C^\alpha_{M_{24}}(g)\to C^\alpha_{M_{24}}(g)$ such that $\varphi(g)=Q$, $\varphi(Q)=g$, and

$$\sum_{b=0}^{N-1}\frac{e^{\frac{2\pi i rb}{N}}}{N}\phi_{{g_{\alpha}}^m,{g_{\alpha}}^{-b}\varphi(k)}(\tau,z)=\sum_{s=0}^{N-1}\frac{e^{\frac{2\pi i ms}{N}}}{N}\phi_{g^r_\alpha,g^{-s}_\alpha k}(\tau,z).$$

This conjecture predicts an outer relabeling symmetry of the central extension that exchanges $g$ and $Q$ and makes the corresponding twisted-sector identifications equivariant; the source gives no resolution status.