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.