Groups with bounded centralizer chains and the Borovik-Khukhro conjecture

Let G be a locally finite group and let F.G/ be the Hirsch-Plotkin radical of G. Let S denote the full inverse image of the generalized Fitting subgroup of G=F.G/ in G. Assume that there is a number k such that the length of every nested chain of centralizers in G does not exceed k. The Borovik-Khukhro conjecture states, in particular, that under this assumption, the quotient G=S contains an abelian subgroup of finite index bounded in terms of k. We disprove this statement and prove a weak analogue of it.