Quasiconformal universality conjecture for Basilica limit sets

Let $K_i$, for $i\in\{1,2\}$, be Basilica limit sets of geometrically finite conformal dynamical systems, such as limit sets of geometrically finite Kleinian groups or Julia sets of geometrically finite rational maps. A point on the ideal boundary of a complementary component is called parabolic or hyperbolic according to the induced dynamical or Fuchsian-group structure. A homeomorphism between $K_1$ and $K_2$ is type-preserving if it sends parabolic points on every ideal boundary bijectively onto parabolic points. Quasiconformal universality conjecture. If $K_1$ and $K_2$ are homeomorphic via a type-preserving map, then $K_1$ is quasi-symmetric to $K_2$. This extends the stated corollary from the paper to all geometrically finite Basilica limit sets; the authors believe their methods may prove it with suitable combinatorial modifications, but the conjecture remains open.