Volume and total scalar-curvature limit conjecture for vortex moduli spaces

Let $\mathcal{H}_\xi$ be the space of holomorphic maps associated with parameter $\xi$, and let $\mathcal{M}_\xi$ be the corresponding vortex moduli space. Assume that $\mathcal{H}_\xi$ embeds as an open dense subset of $\mathcal{M}_\xi$. Denote their metrics by $\omega_{\mathcal{H}}$ and $\omega_{\mathcal{M}}$, respectively, and write $\operatorname{Vol}$ for volume and the total scalar curvature for the integral of scalar curvature against the corresponding volume form. Volume and scalar-curvature limit conjecture. Then

$$\operatorname{Vol}(\mathcal{H}_\xi,\omega_{\mathcal{H}})=\lim_{e^2\rightarrow+\infty}\operatorname{Vol}(\mathcal{M}_\xi,\omega_{\mathcal{M}}).$$

Moreover, the total scalar curvature of $(\mathcal{H}_\xi,\omega_{\mathcal{H}})$ is the $e^2\rightarrow\infty$ limit of the total scalar curvature of $(\mathcal{M}_\xi,\omega_{\mathcal{M}})$. This proposal follows by exchanging the strong-coupling limit with the volume integral and is conditional on the metric-convergence conjecture; its validity is not established in the source.