Almost simple groups polynomial bound conjecture

Let $X$ be a finite almost simple group, meaning that its socle $\operatorname{soc} X$ is a nonabelian simple group and $\operatorname{soc} X\leq X\leq \operatorname{Aut}(\operatorname{soc} X)$. For a normal subgroup $N$ of a finite group $G$, define

$$m(G,N):=m(G)-m(G/N),$$

where $m(G)$ is the largest size of a minimal generating set. Let $\pi(S)$ denote the set of prime divisors of $|S|$. Almost simple groups polynomial bound conjecture. There exist constants $\sigma$ and $\eta$ such that, for every finite almost simple group $X$,

$$m(X,\operatorname{soc} X)\leq \sigma\cdot |\pi(\operatorname{soc} X)|^\eta.$$

The paper presents this as the almost-simple reduction of the polynomial bound conjecture; it is known for alternating and sporadic socles, while the Lie-type case remains substantially open.