Generic simplicity conjecture for Dirichlet complexes of hyperbolic groups

Let $\Gamma<PO(n,1)$ be a discrete subgroup of class $\mathcal K$, and let $x\in\mathbb H^n$. The Dirichlet complex $\mathcal D_x(\Gamma)$ is the Dirichlet complex associated with $\Gamma$ and $x$.

Generic simplicity conjecture. For generic $x\in\mathbb H^n$, the Dirichlet complex $\mathcal D_x(\Gamma)$ is simple.

The conjecture is posed as a consequence of a positive answer to the preceding linear-algebra problem about singular tuples in $O(n,1)^\uparrow$. The source supplies no evidence that it has been resolved.