The quasiconformal rigidity conjecture

Let ${\cal F}$ be a family of maps, and let $f\in{\cal F}$ be non-hyperbolic. The map $f$ is quasiconformally rigid when it is not quasiconformally conjugate to any other map in a sufficiently small neighborhood of $f$ inside ${\cal F}$. Quasiconformal rigidity conjecture. Every non-hyperbolic $f\in{\cal F}$ is quasiconformally rigid. The conjecture concerns local uniqueness in quasiconformal classes; the survey does not report a resolution in this generality.