Polynomial bound conjecture for maximal independent generating sets

Let $G$ be a finite group, let $m(G)$ be the largest size of a minimal generating set of $G$, and let

$$\delta(G):=\sum_{p\in \pi(G)}d_p(G),$$

where $\pi(G)$ is the set of prime divisors of $|G|$ and $d_p(G)$ is the minimal number of generators of a Sylow $p$-subgroup of $G$. Polynomial bound conjecture. There exist constants $c$ and $\eta$ such that, for every finite group $G$,

$$m(G)\leq c\cdot \delta(G)^\eta.$$

This generalizes Dennis' conjecture and is motivated by the polynomial bound established for symmetric groups; the paper reduces it to a corresponding assertion for finite almost simple groups.