Conjecture distinguishing connected Julia sets from Kleinian limit sets

Let $\mathcal J$ be the Julia set of a rational map and let $\Lambda$ be the limit set of a Kleinian group. Suppose that $\mathcal J$ and $\Lambda$ are connected and neither is homeomorphic to the circle or the $2$-sphere. Julia–Kleinian non-equivalence conjecture. There exists no quasiconformal homeomorphism of the Riemann sphere that maps $\mathcal J$ onto $\Lambda$. This conjecture concerns the quasiconformal classification of fractal sets arising in conformal dynamics and Kleinian group theory; the source attributes it to LLMM19 and provides no resolution here.