The folklore polynomial diameter conjecture for transitive permutation groups

Let $G\leq S_n$ be a transitive permutation group. The diameter of $G$ is defined using the maximum over generating sets, namely $\operatorname{diam}(G)=\max\{\operatorname{diam}(G,X):X\text{ a generating set for }G\}$. Folklore conjecture.

$$\operatorname{diam}(G)\ll n^{O(1)}.$$

This conjecture concerns polynomial diameter bounds for arbitrary transitive permutation groups. It is attributed to folklore and was made explicit in work of Kornhauser–Miller–Spirakis and McKenzie. The paper notes that it would follow conditionally from Babai’s conjecture and that no transitive permutation group is known with diameter greater than quadratic.