Weak form of Gaiotto–Moore–Neitzke's conjecture for the $SU(n)$ Hitchin moduli space

Let $({\overline{\partial}}_E,\varphi)$ be a Higgs bundle in the regular locus $\mathcal M'$, and let $g_{L^2}$ and $g_{\mathrm{sf}}$ denote Hitchin's $L^2$-metric and the semiflat metric. Let $Z_{\gamma_0}$ be the period associated with the shortest relevant nonzero charge; in the $SU(2)$ case, write $|Z_{\gamma_0}|=2M$, where $M$ is the length of a shortest geodesic on the spectral cover $\Sigma$, measured in the singular flat metric $\pi^*|\det\varphi|$. Weak form of Gaiotto–Moore–Neitzke's conjecture. Fix a Higgs bundle $({\overline{\partial}}_E,\varphi)$ in $\mathcal M'$. Hitchin's $L^2$-metric admits the expansion

$$g_{L^2}=g_{\mathrm{sf}}+O\left({\mathrm e}^{-2|Z_{\gamma_0}|t}\right).$$

This is the expected leading exponential asymptotic of the Hitchin metric as the scaling parameter $t$ tends to infinity; the displayed claim is explicitly identified as a weak form, while the stronger integral-relation conjecture remains unresolved.