The topological rigidity conjecture

Let ${\cal F}$ be a family of maps, and let $f\in{\cal F}$ be non-hyperbolic. The map $f$ is topologically rigid when it is not topologically conjugate to any other map in a sufficiently small neighborhood of $f$ inside ${\cal F}$. Topological rigidity conjecture. Every non-hyperbolic $f\in{\cal F}$ is topologically rigid. The conjecture is stated for general families considered in the survey and is known in the quadratic-polynomial setting in the cited work; its full scope remains open.