Generic simplicity conjecture for Dirichlet tilings in hyperbolic 3-space

Let $\Gamma<PO(3,1)$ be a discrete subgroup of class $\mathcal K$, and let $x\in\mathbb H^3$ be a point not fixed by any nontrivial element of $\Gamma$. The Dirichlet tiling $\mathcal D_x$ is the Voronoi tiling associated with the orbit $\Gamma\cdot x$.

Generic simplicity conjecture. For generic choice of $x$, the tiling $\mathcal D_x$ is simple, meaning that its dual cell-complex is a simplicial complex.

Simplicity is equivalent to transversality of the bisectors meeting at every boundary point of a Dirichlet domain. The claim is presented as a genericity result needed in the construction, but no resolution status is supplied in the source.