  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 $G/F(G)$ 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 $G/S$ contains an abelian subgroup of index bounded in terms of
$k$. We disprove this statement and prove some its weaker analog.
