Babai's thickness conjecture for primitive coherent configurations

Let $\mathfrak{X}$ be a primitive coherent configuration on $n$ vertices. For a group $G$, its thickness $\theta(G)$ is the greatest $t$ such that the alternating group $A_t$ is involved in $G$. A Cameron scheme is an orbital configuration of a Cameron group. Babai's thickness conjecture. For every $\varepsilon>0$, there is some $N_{\varepsilon}$ such that if $n>N_\varepsilon$ and

$$\theta(\operatorname{Aut}(\mathfrak{X}))\geq n^{\varepsilon},$$

then $\mathfrak{X}$ is a Cameron scheme. This is a thickness formulation of the conjectured classification by large automorphism groups. The paper presents it as open; it is related to bounds connecting thickness, group order, and minimal degree.