Uniform Fitting-height conjecture for fixed-point-free automorphisms satisfying polynomial identities

Let $f(x)\in\mathbb{Z}[x]\setminus\{0\}$. An automorphism $\alpha$ of a finite group $G$ has ordered identity $f(x)$ if the corresponding identity in the group ring holds for $\alpha$; it is fixed-point-free if $C_G(\alpha)=1$. The Fitting height of a finite solvable group is the length of its Fitting series.

Uniform Fitting-height conjecture. For every $f(x)\in\mathbb{Z}[x]\setminus\{0\}$, there exist integers $k(f(x))>0$ and $m(f(x))$ such that, whenever $G$ is a finite solvable group admitting a fixed-point-free automorphism $\alpha$ for which $f(x)$ is an ordered identity, the following hold: (a) the Fitting height of $G$ is at most $m(f(x))$; and (b) if $\gcd(|G|,k(f(x)))=1$, then the Fitting height of $G$ is at most the number of irreducible factors of $f(x)$.

The conjecture seeks a uniform bound depending only on the polynomial identity, with a sharper bound after excluding finitely many primes from $|G|$. It is motivated by results of Thompson, Berger, Turull and others on Fitting height under fixed-point-free automorphisms, but the stated uniform bounds are presented here as a conjecture.