Orbit-closure dichotomy for Apollonian circle packings

Let $\mathcal P$ be an Apollonian circle packing with symmetry group $\Gamma<\operatorname{PSL}_2(\mathbb C)$. For any circle $C\in \mathcal C_\Lambda^*$, consider its orbit $\Gamma C$. Orbit-closure dichotomy. The orbit $\Gamma C$ is either closed or dense in $\mathcal C_\Lambda$. This conjecture extends the orbit-closure classification known for geometrically finite acylindrical groups to the Apollonian group, which is not acylindrical because its compact core is a genus-two handlebody with compressible boundary. Whether the asserted dichotomy holds remains open.