15.37 (2002)

Let $G$ be a group satisfying the minimum condition for centralizers. Suppose that $X$ is a normal subset of $G$ such that $[x, y, \dots, y] = 1$ for any $x, y \in X$ with $y$ repeated $f(x, y)$ times. Does $X$ generate a locally nilpotent (whence hypercentral) subgroup?
If the numbers $f(x, y)$ can be bounded, the answer is affirmative (F. O. Wagner, J. Algebra, 217, no. 2 (1999), 448–460).