Guichard-Wienhard equivalence for higher Teichmüller spaces

Let $S$ be a closed surface and let $G$ be a simple Lie group of higher rank. A connected component of $\operatorname{Hom}(\pi_1(S),G)/G$ is a higher Teichmüller space when it consists entirely of Anosov representations. Characteristic invariants are the usual topological invariants of the relevant associated bundles.

Guichard-Wienhard equivalence. There are connected components of $\operatorname{Hom}(\pi_1(S),G)/G$ which are not distinguished by characteristic invariants if and only if there exist higher Teichmüller spaces in $\operatorname{Hom}(\pi_1(S),G)/G$.

This is presented as a consequence of the preceding conjecture, using that Anosov representations are discrete and injective. The source does not state that the equivalence has been proved.