Contact-tree conjecture for geometrically finite Bers boundary groups

Let $G$ be a geometrically finite Bers boundary group, and let $\Lambda(G)$ be its limit set. Form the contact tree whose vertices are the components of the domain of discontinuity minus the distinguished invariant set, with edges joining components that touch, and color each vertex according to whether the corresponding component is purely accidental parabolic. Equip the tree with its ribbon structure, and call the resulting object the bi-colored contact tree. Bi-colored contact-tree conjecture. The bi-colored contact tree is a complete quasiconformal invariant: two Basilica limit sets arising as limit sets of geometrically finite Bers boundary groups are quasiconformally equivalent if and only if their bi-colored contact trees are isomorphic. The source derives this formulation from the preceding universality conjecture in the Bers-boundary setting; it remains open.