Compact groups in which commutators have finite right Engel sinks

A right Engel sink of an element $g$ of a group $G$ is a subset containing all sufficiently long commutators $[...[[g,x],x],\dots ,x]$. We prove that if $G$ is a compact group in which, for some $k$, every commutator $[...[g_1,g_2],\dots ,g_k]$ has a finite right Engel sink, then $G$ has a locally nilpotent open subgroup. If in addition, for some positive integer $m$, every commutator $[...[g_1,g_2],\dots ,g_k]$ has a right Engel sink of cardinality at most $m$, then $G$ has a locally nilpotent subgroup of finite index bounded in terms of $m$ only.