Simplicity conjecture for cyclic loxodromic Dirichlet tilings

Let $\langle A\rangle<SO(3,1)^\uparrow\cong PSL(2,\mathbb C)$ be a cyclic loxodromic group. Its Dirichlet tiling $\mathcal D_x$ in $\mathbb H^3$ is simple for every $x\in\mathbb H^3$.

Cyclic loxodromic simplicity conjecture. The same conclusion holds for all cyclic loxodromic subgroups of $PO(n,1)$ with $n\geq 3$.

The preceding three-dimensional statement is given as a theorem, attributed to work of T. Drumm and J. Poritz. The conjectural extension concerns all dimensions $n\geq3$ and is not accompanied by a resolution status in the source.