Classification conjecture for GHC-regular representations in anti-de Sitter space

Let $\rho: \Gamma\to\operatorname{SO}_{0}(2,n)$ be a GHC-regular representation. A representation is quasi-Fuchsian if it is a deformation of the inclusion of a lattice in $\operatorname{SO}_{0}(1,n)$, and let $p,q\geq1$ satisfy $p+q=n$. Classification conjecture. Every GHC-regular representation into $\operatorname{SO}_{0}(2,n)$ is either a quasi-Fuchsian representation of a lattice in $\operatorname{SO}_{0}(1,n)$, or a representation of a lattice in $\operatorname{SO}_{0}(1,p)\times\operatorname{SO}_{0}(1,q)$. The conjecture proposes that these are the only types of GHC-regular representations, extending the known constructions of quasi-Fuchsian and product-lattice examples; the general classification remains open.