A remark on the rigidity of the first conformal Steklov eigenvalue

We show rigidity of the first conformal Steklov eigenvalue on annuli and M"obius bands. The proof relies, among others, on uniqueness results due to Fraser–Schoen, a compactness theorem of the second named author, and recent work of the authors on asymptotic control of Steklov eigenvalues in glueing constructions.