Quasiconformal universality conjecture for conformal-dimension-one limit sets

Let $K_i$, for $i\in\{1,2\}$, be limit sets of hyperbolic conformal dynamical systems, such as limit sets of convex co-compact Kleinian groups or Julia sets of hyperbolic rational maps. Suppose that $K_1$ and $K_2$ have conformal dimension $1$ and are homeomorphic. Quasiconformal universality conjecture. Then $K_1$ is quasi-symmetric to $K_2$. More generally, let $K_i$ be 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. If $K_1$ and $K_2$ have conformal dimension $1$ and are homeomorphic via a type-preserving map, then $K_1$ is quasi-symmetric to $K_2$. The conjecture proposes universality for conformal-dimension-one limit sets, extending known rigidity and universality phenomena; the source gives no resolution.