The adiabatic limit conjecture for Hamiltonian vortices

Let $\mu:M\to\mathfrak{g}$ be proper, let $0$ be a regular value of $\mu$, assume that $\mu^{-1}(0)$ is nonempty, and suppose that $\mathrm{G}$ acts freely on $\mu^{-1}(0)$. For $\bar B\in H_2(M/\mskip-6mu/\mathrm{G};\mathbb{Z})$ and $\alpha_1,\dots,\alpha_k\in H^*_{\mathrm{G}}(M;\mathbb{Z})$, let $B\in H^{\mathrm{G}}_2(M;\mathbb{Z})$ be the image of $\bar B$ under the homomorphism induced by $\mu^{-1}(0)\hookrightarrow M$, and let $\bar\alpha_i\in H^*(M/\mskip-6mu/\mathrm{G};\mathbb{Z})$ be the image of $\alpha_i$ under the corresponding homomorphism. Adiabatic limit conjecture. One has

$$\Phi_{B,\Sigma,k}^{M,\mu}(1,\alpha_1,\dots,\alpha_k)={\rm GW}_{\bar B,\Sigma,k}^{M/\mskip-6mu/\mathrm{G}}(\bar\alpha_1,\dots,\bar\alpha_k).$$

Here $1\in H^0(\mathcal{A}/\mathcal{G})$. The conjecture identifies the Hamiltonian gauge-theoretic invariants with Gromov--Witten invariants of the Marsden--Weinstein quotient in the adiabatic limit; the source indicates that it was expected to follow from the correspondence between solutions for sufficiently small adiabatic parameter and holomorphic curves in the quotient, at least when the quotient is semi-positive or weakly monotone.