Labourie's uniqueness conjecture for Hitchin representations

Let $\Gamma$ be the fundamental group of a closed oriented surface $\Sigma$ of genus at least two, and let $\Teich(\Sigma)$ denote the Teichmüller space of marked complex structures on $\Sigma$. Let $G$ be a real split Lie group, let $K$ be a maximal compact subgroup of $G$, and let $\rho$ be a Hitchin representation of $\Gamma$ into $G$. For $J\in\Teich(\Sigma)$, consider the $\rho$-equivariant harmonic map from $(\widetilde{\Sigma},J)$ to the Riemannian symmetric space $G/K$.

Labourie's conjecture. There is a unique complex structure $J\in\Teich(\Sigma)$ on $\Sigma$ such that the $\rho$-equivariant harmonic map from $(\widetilde{\Sigma},J)$ to $G/K$ is weakly conformal.

This conjecture seeks a canonical complex structure associated with a Hitchin representation, removing the dependence of non-abelian Hodge-theoretic parameterizations on the chosen complex structure and restoring a natural mapping class group perspective. Labourie proved that the energy functional is proper for Anosov representations, so a critical point exists; the uniqueness asserted here is not established in the supplied text.