Strong-coupling convergence conjecture for abelian vortex moduli metrics

Let $M$ be a Kähler manifold, let $X$ be a toric target, and let $\mathcal{H}_\xi$ be the space of holomorphic maps $M\rightarrow X$ associated with parameter $\xi$. Let $\mathcal{M}_\xi$ be the corresponding vortex moduli space, equipped with its $L^2$-metric, and use the embedding $\mathcal{H}_\xi\hookrightarrow\mathcal{M}_\xi$. The space $\mathcal{H}_\xi$ has its natural $L^2$-metric determined by the metric on $M$ and the metric on $X$. Strong-coupling convergence conjecture. The metric induced on $\mathcal{H}_\xi$ from the vortex metric on $\mathcal{M}_\xi$ converges pointwise to the natural $L^2$-metric on $\mathcal{H}_\xi$ as the gauge coupling satisfies $e^2\rightarrow\infty$. This conjecture concerns the comparison between vortex moduli metrics and the mapping-space metric in the strong-coupling limit; it has been proved for maps from Riemann surfaces to projective space, but the higher-dimensional case remains open.