Regularity extensions of Pyber, Cameron and Vdovin's conjectures

Let $G$ be a finite group with a core-free maximal subgroup, and let $m(G)$ be the minimal index of a core-free maximal subgroup. Let $R_{\rm \max}(G)$, $R_{\rm ns}(G)$ and $R_{\rm sol}(G)$ be the corresponding regularity invariants. Regularity conjectures.

$$R_{\rm \max}(G)\leqslant c\log_{m(G)}|G|$$

for an absolute constant $c$; $R_{\rm ns}(G)\leqslant 7$ for every finite almost simple group $G$, with equality if and only if $G={\rm M}_{24}$; and $R_{\rm sol}(G)\leqslant 5$ for every finite group $G$ with trivial soluble radical. These are proposed natural extensions of the preceding base-size conjectures; the supplied text records partial results and special cases, but does not establish resolution of the full three-part proposal.