Topological invariants conjecture for maximal representations

Let $G$ be a simple Lie group of Hermitian type, and let $\operatorname{Rep}_{\mathrm{max}}(\pi_1(\Sigma),G)$ denote the maximal representation space. The topological invariants of Theorem~ are associated to these representations.

Topological invariants conjecture. If $G$ is not locally isomorphic to $\mathrm{Sp}(2n,\mathbf{R})$, then the topological invariants of Theorem~ distinguish connected components of $\operatorname{Rep}_{\mathrm{max}}(\pi_1(\Sigma),G)$.

For maximal representations into simple Hermitian groups other than the symplectic groups, this predicts that the available topological invariants completely determine the connected component. The source presents this as a conjecture; no resolution is supplied here.