The residually finite quasi-Engel group conjecture

Let $G$ be a quasi-Engel group, meaning that for the fixed initial word $w$ there is an integer $n$ such that the identity ${}^w\!u_n(x,y)\equiv 1$ holds in $G$. A group is residually finite if the intersection of all its normal subgroups of finite index is trivial. A property holds locally if it holds for every finitely generated subgroup. The residually finite quasi-Engel group conjecture. Every residually finite, quasi-Engel group is locally solvable. The question concerns whether a bounded quasi-Engel identity forces local solvability under residual finiteness; the corresponding Engel-group problem is stated as open in the source, while positive results are known in several related settings.