Goldman–Wentworth–Labourie uniqueness conjecture for conformal structures

Let $\Sigma$ be a surface, let $G^\tau$ be a split real simple Lie group, and let $\theta$ be a representation in the Hitchin component. For a conformal structure on $\Sigma$, let $(E,\Phi)$ be the associated Higgs bundle, let $k$ be the Killing form, and write $q_2=k(\Phi,\Phi)$ for its holomorphic quadratic differential. Goldman–Wentworth–Labourie conjecture. Given a representation $\theta$ in the Hitchin component, there is a unique conformal structure on $\Sigma$ such that $(E,\Phi)$ satisfies

$$q_2=k(\Phi,\Phi)=0.$$

Since $q_2$ vanishes exactly when the associated equivariant harmonic map is conformal, the conjecture asserts uniqueness of the conformal structure making that harmonic map conformal. It is proved for the Hitchin component of ${\rm PSL}(2,\mathbb{R})$, while the general case is presented as open here.