The self-exponential formulation of Generalized Moonshine

Let $V^\natural$ be the Moonshine vertex operator algebra, let $\mathbb{M}$ be the Monster group, and let $\alpha^\natural\in H^3(\mathbb{M},U(1))$ be the Moonshine anomaly. Write $\operatorname{Ell}^{\alpha^\natural}(B\mathbb{M})$ for the corresponding twisted equivariant elliptic cohomology group. Self-exponential Generalized Moonshine conjecture. $V^\natural$ defines an element of

$$\operatorname{Ell}^{\alpha^\natural}(B\mathbb{M})$$

that is self-exponential. The source says this remains conjectural because “defines” and “self-exponential” are not yet precisely defined.