Conjecture on finite-covolume actions of discrete groups on hyperbolic isometry groups

Let $n\geq 2$. Let $\Gamma$ be a discrete group and let $j\times\rho$ be a faithful representation of $\Gamma$ into $\operatorname{SO}_0(n,1)\times\operatorname{SO}_0(n,1)$. Suppose that $(j\times\rho)(\Gamma)$ acts properly discontinuously on $\operatorname{SO}_0(n,1)$ with finite covolume. Finite-covolume action conjecture. Then, up to switching the factors, $j$ is discrete and faithful and $j(\Gamma)$ is a lattice in $\operatorname{SO}_0(n,1)$; in particular, it is geometrically finite. This conjecture would imply the preceding geometric-finiteness expectation and is posed as a broader question about finite-covolume proper actions.