Groups with bounded centralizer chains an 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.