The bounded exponent conjecture for fixed points of coprime operator groups

Let $q$ be a prime, $m$ a positive integer, and let $A$ be an elementary abelian group of order $q^r$, with $r\ge 2$, acting on a finite $q'$-group $G$. For a group $H$, let $\gamma_i(H)$ denote the $i$th term of its lower central series, and let $H^{(i)}$ denote the $i$th term of its derived series. Here $A^\#$ denotes the set of nontrivial elements of $A$.

Bounded exponent conjecture. If $\gamma_{r-1}(C_G(a))$ has exponent dividing $m$ for every $a\in A^\#$, then $\gamma_{r-1}(G)$ has $\{m,q,r\}$-bounded exponent. Moreover, if $d$ is an integer satisfying $2^d\le r-1$ and the $d$th derived group of $C_G(a)$ has exponent dividing $m$ for every $a\in A^\#$, then the $d$th derived group $G^{(d)}$ has $\{m,q,r\}$-bounded exponent.

This conjecture was proposed as a common generalization of two bounded-exponent theorems for coprime operator groups. It predicts that suitable exponent bounds on fixed-point subgroups force corresponding bounds on the lower-central or derived terms of the whole group, with dependence only on $m$, $q$, and $r$.