Witten's localization formula for reductions at regular co-adjoint orbits

For each $t\in\mathbb{R}$, $X\in\mathfrak{g}$, let $\Theta(M,t)(X):=\int_M e^{-itd_X\lambda^{M_{\mathcal O}}}\alpha(X)$, and let $\Theta_0^{\mathcal O}$ denote the generalized-function limit of $\Theta(M_0^{\mathcal O},t)$ as $t\to\infty$. Let $M^{\mathcal O}_{\mathrm{red}}=\mu^{-1}(\mathcal O)/G$ be the reduction at the regular co-adjoint orbit $\mathcal O$, and let $W:C^{\infty}(\mathfrak{g})^G\to H^*(M^{\mathcal O}_{\mathrm{red}})$ be the Chern–Weil homomorphism associated to the principal fibration $\mu^{-1}(\mathcal O)\to M^{\mathcal O}_{\mathrm{red}}$. For a test function $\Phi$, Witten conjectured that

$$\int_{\mathfrak g}\Theta_0^{\mathcal O}(X)\Phi(X)dX=(2\pi i)^{\dim G}\operatorname{vol}(G)\int_{M^{\mathcal O}_{\mathrm{red}}}\alpha^{\mathcal O}_{\mathrm{red}}W(\Phi).$$

This formula is intended to express the contribution from the reduced space in the Jeffrey–Kirwan–Witten localization framework, relating equivariant integration to the Chern–Weil image on the symplectic reduction. The supplied text identifies it as Witten's conjecture but gives no resolution status.