Fitting-series containment conjecture under centralizing fixed points

Let $G$ be a solvable group and let $A$ act on $G$ by automorphisms with $(|A|,|G|)=1$. Let $F(G)$ be the Fitting subgroup, define $F_0(G)=1$, $F_1(G)=F(G)$, and let $F_{i+1}(G)$ be the full inverse image of $F(G/F_i(G))$ in $G$. Let $P$ be an $A$-invariant Sylow $p$-subgroup of $G$ such that $p\notin\pi(C_G(A))$, and suppose that $C_G(A)$ normalizes and centralizes $P$. Let $l(A)$ be the number of prime factors of $|A|$, counted with multiplicity. Centralizing fixed-points conjecture. $P$ is contained in $F_n(G)$ where $n=l(A)$. This weaker conjecture is stated as a consequence of the first conjecture and would imply the well-known conjecture that the Fitting height of $G$ is bounded by $l(A)$ when $C_G(A)=1$; the supplied text does not indicate whether it is proved or refuted.