Local removability conjecture

Let $E\subset\mathbb{C}$ be compact. It is conformally removable if every homeomorphism of $\widehat{\mathbb{C}}$ that is conformal outside $E$ is conformal everywhere. It is locally conformally removable if, for every open set $U$, every homeomorphism on $U$ that is conformal on $U\setminus E$ is conformal on all of $U$.

Local removability conjecture. A compact set is conformally removable if and only if it is locally conformally removable.

Local conformal removability implies conformal removability, while whether the converse holds is unknown. The question is also related to a question of Bishop.