Naive localization formula for the index on the moduli stack

Let $G$ be a rank-one compact Lie group with maximal torus $T$, let $\mathfrak{M}$ be the moduli stack of $G$-bundles, and let $\mathfrak{M}_T$ be the corresponding moduli stack of $T$-bundles. Let $\nu^*$ be the equivariant $K$-theory conormal bundle to the morphism $\mathfrak{M}_T\to\mathfrak{M}$, and let an admissible class mean a class for which the indicated indices are defined. The naive localization conjecture. The index of an admissible class over $\mathfrak{M}$ is one-half the index of its restriction to $\mathfrak{M}_T$, divided by the equivariant $K$-theory Euler class of $\nu^*$:

$$\operatorname{Ind}_{\mathfrak{M}}(E)=\frac{1}{2}\,\frac{\operatorname{Ind}_{\mathfrak{M}_T}(E|_{\mathfrak{M}_T})}{\operatorname{Euler}_{K}(\nu^*)}.$$

This is a conjectural localization formula relating the index on the nonabelian moduli stack to the torus moduli stack. The source gives no resolution, so the conjecture is recorded as open.