On groups in which Engel sinks are cyclic

For an element $g$ of a group $G$, an Engel sink is a subset $\mathcal{E}(g)$ such that for every $x\in G$ all sufficiently long commutators $[x,g,g,\ldots,g]$ belong to $\mathcal{E}(g)$. We conjecture that if $G$ is a profinite group in which every element admits a sink that is a procyclic subgroup, then $G$ is procyclic-by-(locally nilpotent). We prove the conjecture in two cases -- when $G$ is a finite group, or a soluble pro-$p$ group.