Pyber's base size conjecture for primitive permutation groups

Let $G$ be a finite primitive permutation group on a finite set $S$. A base is a subset $T\subseteq S$ satisfying $G_T=\{e\}$, and $b(G)$ denotes the minimum cardinality of a base. Pyber's base size conjecture.

$$b(G)=\Theta\left(\frac{\log|G|}{\log|S|}\right).$$

The orbit-stabilizer theorem gives the lower bound $b(G)\geq \log|G|/\log|S|$, so the conjecture asserts that this bound is asymptotically tight for primitive actions. The conjecture was announced as proved by Duyan, Halasi, and Maróti; the almost simple case was already verified by Burness.