Goldman's volume rigidity conjecture for uniform lattices

Let $\Gamma$ be a uniform lattice in a connected semisimple Lie group $G$ with trivial center and no compact factors. Let $X$ be the associated symmetric space, let $M=\Gamma\backslash X$, and let $\rho\colon \Gamma\to G$ be a representation. Its volume invariant satisfies

$$|\upsilon(\rho)|\leq \operatorname{Vol}(M).$$

Goldman's volume rigidity conjecture. Equality holds if and only if $\rho$ is a discrete, faithful representation of $\Gamma$ into $G$. Goldman proved this conjecture for all connected semisimple Lie groups except $\mathrm{SU}(n,1)$, $\mathrm{Sp}(n,1)$, and $\mathrm{F}_4^{-20}$; the source presents the equality characterization as conjectural in the remaining cases.