The weakly-top group abelianization growth conjecture

Let $G$ be a finite weakly-top group, and write $G'$ for its derived subgroup. The quotient $G/G'$ is the abelianization of $G$.

Weakly-top group abelianization growth conjecture. There exists $\delta>0$ such that every weakly-top group $G$ satisfies

$$|G/G'| \geq |G|^\delta.$$

The question arises from the possibility that weakly-top groups have bounded derived length; the examples discussed in the paper show that $\delta$ cannot be larger than $1/2$. The conjecture is presented as an open problem, and no resolution is supplied here.