Dumas–Sanders fiber-bundle conjecture for Kapovich–Leeb–Porti manifolds

Let $S$ be the closed surface underlying the quasi-Hitchin representations under consideration, let $M_{\mathbb C}$ be the manifold associated with a quasi-Hitchin representation in a complex semisimple group, and let the other manifolds be those obtained from such a representation by the Kapovich–Leeb–Porti construction. Dumas–Sanders conjecture. The manifold $M_{\mathbb C}$, and the other manifolds obtained in this way, is homeomorphic to a continuous fiber bundle over $S$. This conjecture concerns the topology of manifolds constructed from quasi-Hitchin representations; the source reports that the homology ring of $M_{\mathbb C}$ and related manifolds has been computed, but does not state that the fiber-bundle description is proved.