The strong conciseness conjecture for profinite groups

Let $w(x_1,\dots,x_k)$ be a word in $k$ variables, and let $G$ be a profinite group. Write $w\{G\}$ for the set of word-values of $w$ in $G$, and $w(G)=\langle w\{G\}\rangle$ for the verbal subgroup. A word is strongly concise in $G$ if $|w\{G\}|<2^{\aleph_0}$ implies that $w(G)$ is finite. Strong conciseness conjecture. Every word is strongly concise in the class of profinite groups. This strengthens ordinary conciseness by replacing finiteness of the word-value set with cardinality strictly below the continuum; the supplied text does not state whether this conjecture has been resolved.