Nondegeneracy conjecture for the energy minimum of Hitchin representations

Let $S$ be a closed oriented surface, let $\rho$ be a Hitchin representation, and let $e_\rho$ denote its energy functional on Teichmüller space. Nondegeneracy conjecture. The minimum of $e_\rho$ is non-degenerate. This is known for $n=2$ and can be proved for $n=3$ using the relationship with real projective structures and affine spheres; in general, the conjecture would imply that the Hitchin map is a homeomorphism.