He-Schramm rigidity conjecture: circle domains with conformally removable boundary are rigid

Claims that every circle domain whose boundary is conformally removable is conformally rigid, with no bound on the number of complementary components and no boundary-extension hypothesis on the map. An intermediate theorem shows that a function continuous across a compact totally disconnected removable set with finite Dirichlet energy off it is globally Sobolev. Together with Rajala's counterexample this settles both directions of the He-Schramm equivalence: removable implies rigid, the converse fails. It does NOT characterize rigid circle domains.

Claims that every circle domain whose boundary is conformally removable is conformally rigid, with no bound on the number of complementary components and no boundary-extension hypothesis on the map. An intermediate theorem shows that a function continuous across a compact totally disconnected removable set with finite Dirichlet energy off it is globally Sobolev. Together with Rajala's counterexample this settles both directions of the He-Schramm equivalence: removable implies rigid, the converse fails. It does NOT characterize rigid circle domains.