Guichard-Wienhard geometric-structure conjecture for higher Teichmüller representations

Let $S$ be a closed surface, let $G$ be a simple Lie group, and let $\rho:\pi_1(S)\to G$ be a representation in a higher Teichmüller space. A generalized flag variety is a homogeneous space $X=G/P$ for a parabolic subgroup $P<G$. Let $M\to S$ be a compact fiber bundle, and let

$$\overline{\rho}:\pi_1(M)\to\pi_1(S)\to G$$

be the representation induced by the bundle map and by $\rho$.

Guichard-Wienhard conjecture. There exists a generalized flag variety $X$ and a compact fiber bundle $M\to S$ such that $\overline{\rho}$ is the holonomy of a locally homogeneous $(G,X)$-structure on $M$.

The conjecture seeks a uniform geometric realization of higher Teichmüller representations. The source gives several examples for Hitchin and maximal representations, but does not establish the assertion in general.