Periodic locally nilpotent groups of finite $c$-dimension

According to Bryant's theorem a periodic locally nilpotent group satisfying minimal condition on centralizers is virtually nilpotent. The $c$-dimension of a group is the supremum of lengths of chains of centralizers. We bound the index of the nilpotent radical of a locally nilpotent $p$-group of finite $c$-dimension $k$ in terms of $k$ and $p$.