Properness and non-degeneracy conjecture for Hitchin-component energy

Let $\Sigma$ be a surface, let $G$ be a split real simple Lie group, and let $\theta$ be a representation in a Hitchin component. Let $E_\theta$ be the associated energy function on Teichmüller space, defined using the $\theta$-equivariant harmonic map. Properness and non-degeneracy conjecture. For a representation in any Hitchin component the associated energy function is proper and all stationary points are non-degenerate minima. This refines the uniqueness conjecture: properness gives existence of stationary points, while the asserted local structure is stronger than uniqueness. The statement is known in the ${\rm PSL}(2,\mathbb{R})$ Hitchin component through Tromba's result, but is otherwise left open in the source.