Borcherds's unified Moonshine conjecture

Let $\mathbb{M}$ be the Monster group. For each $g\in\mathbb{M}$, seek a $\frac{1}{|g|}\mathbb{Z}$-graded super-module ${}^g\hat V$ over $\mathbb{Z}[e^{2\pi i/|g|}]$, equipped with an action of a central extension $(\mathbb{Z}/|g|\mathbb{Z}).C_{\mathbb{M}}(g)$, satisfying all five properties stated in the conjecture: ${}^1\hat V$ is a self-dual integral form of $V^\natural$; commuting $g,h$ satisfy $\hat H^*(\hat g,{}^h\hat V)\cong{}^{gh}\hat V\otimes\mathbb{Z}/|g|\mathbb{Z}$ for a lift $\hat g$; Fricke elements satisfy the two stated Tate-cohomology and complexification properties; ${}^g\hat V$ is “often” a vertex superalgebra with the stated form; and coprime commuting elements have the stated graded Brauer character. Borcherds's unified Moonshine conjecture. Such a rule assigning these modules and satisfying all those properties exists. This would unify Generalized and Modular Moonshine, but the construction and the required integral and mixed-characteristic compatibility remain incomplete.