David hierarchy conjecture for Basilica limit sets

Let $K_i$, for $i\in\{1,2\}$, be Basilica limit sets of geometrically finite conformal dynamical systems. A map is weakly type-preserving if it sends parabolic points to parabolic points, without requiring hyperbolic points to avoid mapping to parabolic points. David hierarchy conjecture. If there is a weakly type-preserving map from $K_1$ to $K_2$, then there exists a David homeomorphism

$$\phi:\widehat{\mathbb{C}}\longrightarrow\widehat{\mathbb{C}}$$

with $\phi(K_1)=K_2$. The conjecture generalizes the proved David-hierarchy result for Kleinian or rational Basilica limit sets and remains open in the stated generality.