Quasisymmetries of Sierpiński carpet Julia sets

We prove that if ξ is a quasisymmetric homeomorphism between Sierpiński carpets that are Julia sets of postcritically-finite rational maps, then ξ is the restriction of a Möbius transformation. This implies that the group of quasisymmetric homeomorphisms of a Sierpiński carpet Julia set of a postcritically-finite rational map is finite.