Finite groups with Engel sinks of bounded rank

For an element $g$ of a group $G$, an Engel sink is a subset ${\mathscr E}(g)$suchthatforevery$x\in G$allsufficientlylongcommutators$[…[[x,g],g],\dots ,g]$belongto${\mathscr E}(g)$. A~finite group is nilpotent if and only if every element has a trivial Engel sink. We prove that if in a finite group $G$ every element has an Engel sink generating a subgroup of rank~$r$, then $G$ has a normal subgroup $N$ of rank bounded in terms of $r$ such that $G\mathrm{/}N$ is nilpotent.