On existence of degenerate circle-preserving maps

Recently, B. Li and Y. Wang proved that if (n⩾2) is a circle-preserving map, then f is a Möbius transformation if and only if f is a non-degenerate map, where a map f is degenerate if the image is a circle. Furthermore, they conjectured that there should exist no degenerate map, or equivalently, f is a Möbius transformation if and only if f is a circle-preserving map. In this note, we construct a degenerate circle-preserving map to show that the conjecture is not true.