Twisted Gysin formula for the index over the moduli stack

Let $G$ be a compact Lie group, let $\Sigma$ be a closed surface of genus $g$, and let $\mathfrak{M}$ be the moduli stack of $G$-bundles on $\Sigma$. Let $\Pi:G^{2g}\to G$ be the product-of-commutators map, let $\tau$ be a transgressed twisting, and let ${}^\tau 1$ be the class defined by the trivialization of $\Pi^*\tau$. Let $\tau_0$, $\sigma$, and $\alpha(V)$ have the meanings in the index construction. The twisted Gysin conjecture.

$$\operatorname{Ind}\left(\mathfrak{M};\mathcal{O}(\tau_0-\sigma)\otimes\exp[t\alpha(V)]\right)={}^\tau Tr\left(\Pi_!{}^\tau 1\right)\in\mathbb{C}[[t]].$$

This is the principal conjectural identification between the index on the moduli stack and the twisted Frobenius trace of the Gysin push-forward. The source gives no resolution, so the conjecture is recorded as open.