The finite-group equivariant VOSA conjecture

Let $\mathbb{V}$ be a unitary self-dual VOSA of central charge $c=n/2$, equivariant under a finite group $G$ with anomaly $k\in (\widetilde{I_{\mathbb{Z}\Omega^{\mathrm{spin}}}})^4(BG)$. Let $\mathbb{W}$ be its canonically $\mathbb{Z}/2$-twisted module, and let $\tilde k$ be the induced twist in the target $\operatorname{KO}$-theory. The finite-group equivariant VOSA conjecture. There is a class

$$[\mathbb{V}]\in \operatorname{TMF}^{n+k}_G(\mathrm{pt})$$

such that

$$\Phi([\mathbb{V}])=\eta(q)^{-n}\mathbb{W}\in \operatorname{KO}^{n+\tilde k}_G((q))(\mathrm{pt}).$$

This extends the affine-equivariant proposal to arbitrary finite symmetry groups and incorporates the anomaly and its induced graded-Brauer twist; the required equivariant twisted $\operatorname{TMF}$ interpretation remains conjectural.