The $\mathbb{F}_1$-module structure conjecture for the integral Moonshine module

Let $V^\natural_{\mathbb{Z}}$ be a self-dual integral form of $V^\natural$ with Monster symmetry, and let $W_\infty M$ denote the big Witt vectors of a graded module $M$. $\mathbb{F}_1$-module structure conjecture. There is an $\mathbb{F}_1$-module structure on $V^\natural_{\mathbb{Z}}$ with $\mathbb{M}$-symmetry, meaning that for each $g\in\mathbb{M}$ there is a distinguished lift of the $(\mathbb{Z}/|g|\mathbb{Z}).C_{\mathbb{M}}(g)$-action on $\hat H^*(g,V^\natural_{\mathbb{Z}})$, as a graded module, to $W_\infty\hat H^*(g,V^\natural_{\mathbb{Z}})$. The source presents this as speculative: such a structure could provide canonical mixed-characteristic lifts, but no construction is known.