He–Schramm rigidity conjecture for circle domains

A circle domain $\Omega\subset\widehat{\mathbb C}$ is a domain whose boundary components are points or circles. It is conformally rigid if every conformal map of $\widehat{\mathbb C}$ that is conformal outside $\partial\Omega$ maps $\Omega$ onto itself, and a set is conformally removable if every homeomorphism of the sphere conformal off that set is Möbius. He–Schramm's rigidity conjecture. A circle domain $\Omega$ is conformally rigid if and only if its boundary $\partial\Omega$ is conformally removable. The conjecture was subsequently disproved by Rajala: the forward implication fails, while the reverse direction remains open.