Anosov density conjecture for compact hyperbolic triangle reflection groups

Let $\Gamma$ be a compact hyperbolic triangle reflection group, and let $\rho\colon \Gamma\rightarrow \operatorname{PGL}(3,\mathbb{R})$ be a representation.

Anosov density conjecture. The representation $\rho$ is discrete and faithful if and only if it lies in the closure of the space of Anosov representations in $\mathrm{Hom}(\Gamma,\operatorname{PGL}(3,\mathbb{R}))$.

The conjecture seeks to characterize all discrete and faithful representations of compact hyperbolic triangle reflection groups into $\operatorname{PGL}(3,\mathbb{R})$ through Anosov representations. The paper proves a classification of the Anosov representations, but the broader characterization of all discrete and faithful representations remains open.