Groups with bounded centralizer chains and the~Borovik--Khukhro conjecture

Let $G$ be a locally finite group and $F(G)$ the Hirsch–Plotkin radical of $G$. Denote by $S$ the full inverse image of the generalized Fitting subgroup of $G\mathrm{/}F(G)$ in $G$. Assume that there is a number $k$ such that the length of every chain of nested centralizers in $G$ does not exceed $k$. The Borovik–Khukhro conjecture states, in particular, that under this assumption the quotient $G\mathrm{/}S$ contains an abelian subgroup of index bounded in terms of $k$. We disprove this statement and prove some its weaker analog.