  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.
