On a conjecture of Simpson

On a compact Riemann surface $\mathrm{\Sigma }$ of genus $g\ge 2$, equipped with a complex vector bundle $E$ of rank $2$ and degree zero let $M_H$ be the moduli space of Higgs bundles. $M_H$ admits a ${\mathbb{C}}^{\star }$-action and to each stable ${\mathbb{C}}^{\star }$-fixed point $[({\bar{\mathrm{\partial }}}_0,{\mathrm{\Phi }}_0)]$ is associated a holomorphic Lagrangian submanifold $W^1({\bar{\mathrm{\partial }}}_0,{\mathrm{\Phi }}_0)$ inside the de Rham moduli space $M_{dR}$ of complex flat connections. In this note we prove a conjecture of Simpson stating that $W^1({\bar{\mathrm{\partial }}}_0,{\mathrm{\Phi }}_0)$ is closed inside $M_{dR}$.