He–Schramm's Cantor-set removability conjecture for circle-domain boundaries

Let $\Omega$ be a circle domain, and let $\partial\Omega$ denote its boundary. A Cantor set is a compact, perfect, totally disconnected set. A set is conformally removable if every homeomorphism of $\widehat{\mathbb{C}}$ that is conformal outside the set is conformal everywhere.

He–Schramm's removability conjecture. If every Cantor set contained in $\partial\Omega$ is conformally removable, then $\partial\Omega$ is conformally removable.

This conjecture concerns the removability of boundaries of circle domains and asks whether removability of all Cantor subsets suffices for removability of the whole boundary. The general question remains open.