Labourie's conjecture on unique equivariant minimal surfaces

Let $G$ be a split real Lie group or a Hermitian Lie group, let $K$ be a maximal compact subgroup of $G$, and let $\rho$ be a Hitchin representation in the split-real case or a maximal representation in the Hermitian case.

Labourie's conjecture. Does there exist a unique $\rho$-equivariant minimal surface in $G/K$?

This generalizes the question of whether Labourie's parametrization by conformal harmonic maps is bijective. Its resolution is not specified in the source.