On quasisymmetry of quasiconformal mappings

Suppose that f:D→D′ is a quasiconformal mapping, where D and D′ are domains in Rn, and that D is a broad domain. We show that for every arcwise connected subset A in D, the weak quasisymmetry of the restriction f|A: A→f(A) implies its quasisymmetry. As a consequence, we see that the answer to one of the open problems raised by Heinonen from 1989 is affirmative, under the additional condition that A is arcwise connected.