Before revision:  Our proof is based on generalizations of the Mabillard--Wagner theorem on construction of almost r-embeddings from equivariant maps, and of the \"Ozaydin theorem on existence of equivariant maps.
After revision:  Our proof is based on generalizations of the Mabillard-Wagner theorem on construction of almost r-embeddings from equivariant maps, and of the \"Ozaydin theorem on existence of equivariant maps.