19.44 (2018)

By definition a profinite group has finite rank at most $r$ if every subgroup of it can be (topologically) generated by $r$ elements. Suppose that for every element $g$ of a profinite group $G$ there is a closed subgroup $E_g$ of finite rank such that for every $x \in G$ all sufficiently long Engel commutators $[x, g, \dots, g]$ belong to $E_g$, that is, for every $x \in G$ there is a positive integer $n(x, g)$ such that $[x, {}_n g] \in E_g$ whenever $g$ is repeated $\geqslant n(x, g)$ times. Is it true that $G$ has a normal subgroup $N$ of finite rank with locally nilpotent quotient $G/N$?