On profinite groups admitting a word with only few values

A group-word w is called concise if the verbal subgroup w(G) is finite whenever w takes only finitely many values in a group G. It is known that there are words that are not concise. The problem whether every word is concise in the class of profinite groups remains wide open. Moreover, there is a conjecture that every word w is strongly concise in profinite groups, that is, w(G) is finite whenever G is a profinite group in which w takes less than \(2^{\aleph _0}\) $2^{{\aleph }_0}$ values. In this paper we show that if the word w takes less than \(2^{\aleph _0}\) $2^{{\aleph }_0}$ values in a profinite group G then w(w(G)) is finite.