On conciseness of words in profinite groups

Let w be a group word. It is conjectured that if w has only countably many values in a profinite group G  , then the verbal subgroup w(G)w(G) is finite. In the present paper we confirm the conjecture in the cases where w   is a multilinear commutator word, or the word x2x2, or the word [x2,y][x2,y].