The symmetry-rigidity conjecture for proper holomorphic maps between balls

Let $f: \operatorname{\mathbb{B}}^m \rightarrow \operatorname{\mathbb{B}}^M$ be a proper holomorphic map, and let $\operatorname{\mathsf{G}}_f$ denote its symmetry group. Suppose that $f$ extends to a map $\overline{\operatorname{\mathbb{B}}^m} \rightarrow \overline{\operatorname{\mathbb{B}}^M}$. Symmetry-rigidity conjecture. If $\operatorname{\mathsf{G}}_f$ is sufficiently large and the extension is sufficiently regular, then, up to composition with automorphisms, $f$ is the map $z \mapsto (z,0)$ from the trivial example. The precise meanings of “sufficiently large” and “sufficiently regular” are left imprecise in the source; later results in the paper establish a precise theorem under stronger symmetry and Hölder regularity assumptions.