A linear derived-length bound for products of conjugacy classes

Let $G$ be a finite solvable group and let $A$ be a conjugacy class of $G$. Write $\eta(AA^{-1})$ for the number of conjugacy classes occurring in the product $AA^{-1}$, and let ${\bf C}_G(A)$ denote the centralizer of the conjugacy class $A$. Conjugacy-class derived-length conjecture. There exist universal constants $q$ and $r$ such that

$$\operatorname{dl}(G/{\bf C}_G(A)) \leq q\eta(AA^{-1})+r.$$

This asks whether the supersolvable bound with constants $q=2$ and $r=-1$ extends, with possibly different universal constants, to all finite solvable groups. It is presented as the conjugacy-class analogue of a known character-theoretic linear bound.