Twisted chiral three-dimensional quantum gravity conjecture for Monster elements

Let $\mathbb M$ be the Monster, let $g\in\mathbb M$, let $\Gamma_g$ be the associated group, and let $\Gamma_{\infty}$ denote the cusp subgroup. Let $T_{\Gamma}^{(1)}$ be the normalized Rademacher sum, and let $V^{\natural}$ and $V^{\natural}_g$ denote the moonshine vertex operator algebra and its simple $g$-twisted module. Twisted chiral quantum-gravity conjecture. There exists a family of twisted chiral three-dimensional quantum gravities at central charge $c=24$, indexed by $g\in\mathbb M$, whose partition functions are naturally given by sums over geometries parameterized by $\Gamma_{\infty}\backslash\Gamma_g$ and coincide with

$$T_{\Gamma}^{(1)}(-1/z),\qquad \Gamma=\Gamma_g.$$

The untwisted member has a VOA structure isomorphic to $V^{\natural}$, and the member corresponding to $g$ has a $g$-twisted $V^{\natural}$-module structure isomorphic to that of $V^{\natural}_g$. This connects monstrous moonshine and three-dimensional quantum gravity; the source presents the claim as its first conjecture, with no resolution supplied.