The profinite unbounded quasi-Engel group conjecture

Let $G$ be an unbounded quasi-Engel group, meaning that for every $x,y\in G$ there is an integer $n=n(x,y)$ such that ${}^w\!u_n(x,y)=1$, for the fixed initial word $w$. A group is profinite if it is a profinite topological group, and a property holds locally if it holds for every finitely generated subgroup. The profinite unbounded quasi-Engel group conjecture. Every profinite, unbounded quasi-Engel group is locally solvable. This is posed as the profinite analogue of the preceding residual-finiteness conjecture. The source records a theorem that every profinite, unbounded Engel group is locally nilpotent, but gives no resolution of the quasi-Engel assertion.