Asymptotics of Hitchin's metric on the Hitchin section

We consider Hitchin's hyperk\"ahler metric $g$ on the moduli space $\mathcal{M}$ of degree zero $\mathrm{SL}(2)$-Higgs bundles over a compact Riemann surface. It has been conjectured that, when one goes to infinity along a generic ray in $\mathcal{M}$, $g$ converges to an explicit "semiflat" metric $g^{\mathrm{sf}}$, with an exponential rate of convergence. We show that this is indeed the case for the restriction of $g$ to the tangent bundle of the Hitchin section $\mathcal{B} \subset \mathcal{M}$.