Higher-twisting Verlinde index formula

Let $G$ be a compact Lie group, let $\mathfrak{M}$ be the moduli stack of $G$-bundles over a genus-$g$ surface, let $\tau_0$ and $\sigma$ be the twistings appearing in the index construction, and let $\alpha(V)$ be the index bundle associated to a representation $V$ of $G$. For $f\in F^{reg}/W$, let $\theta_f(t)$ denote the structure constants of the Frobenius algebra ${}^\tau K_G^{\dim G}(G)\otimes\mathbb{C}[[t]]$ over $\mathbb{C}[[t]]$. The higher-twisting conjecture.

$$\operatorname{Ind}\left(\mathfrak{M};\mathcal{O}(\tau_0-\sigma)\otimes\exp[t\alpha(V)]\right)=\sum_{f\in F^{reg}/W}\theta_f(t)^{1-g}.$$

This is the proposed extension of the index formula to higher twistings, expressing the index as a sum of powers of Frobenius-algebra structure constants. The source gives no resolution, so the conjecture is recorded as open.