The Engel-value conjecture for verbal subgroups of residually finite groups

Let $w$ be a group-word and let $n$ be a positive integer. For a group $G$, write $w(G)$ for the verbal subgroup generated by all values of $w$ in $G$. An element is $n$-Engel if, for every $g\in G$, one has $[g,_{n}x]=1$; assume that every $w$-value in $G$ is $n$-Engel. Engel-value conjecture. If $G$ is residually finite, then the verbal subgroup $w(G)$ is locally nilpotent. This extends the known result for multilinear commutator words and is motivated by the theorem that residually finite $n$-Engel groups are locally nilpotent; the conjecture asks for local nilpotence of the verbal subgroup under the corresponding Engel condition.