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You are right, that was already contained in the statement of the question. Sorry. Then things get much harder. If the set is nice enough, then maybe something like Theorem 5.2 in MR2200122 (2006m:53061) (Lang, Urs; Schlichenmaier, Thilo Nagata dimension, quasisymmetric embeddings, and Lipschitz extensions. Int. Math. Res. Not. 2005, no. 58, 3625–3655.) could help.
