Rigidity of proper holomorphic self-mappings of the pentablock

The pentablock is a Hartogs domain in C3C3 over the symmetrized bidisc in C2C2. The domain is a bounded inhomogeneous pseudoconvex domain, which does not have a C1C1 boundary. Recently, Agler–Lykova–Young constructed a special subgroup of the group of holomorphic automorphisms of the pentablock, and Kosiński fully described the group of holomorphic automorphisms of the pentablock. The aim of the present study is to prove that any proper holomorphic self-mapping of the pentablock must be an automorphism.