19.19 (2018)

A finite transitive permutation group $G$ is said to have the road closure property if, given any orbit $O$ of $G$ on 2-sets, and any proper block of imprimitivity for $G$ acting on $O$, the graph with edge set $O \setminus B$ is connected. Such a group must be primitive, and basic (not contained in a wreath product with the product action); it cannot have an imprimitive subgroup of index 2. In addition, such a group cannot be one of the permutation groups arising from triality (whose socle is $D_4(q)$ and intersects the point stabiliser in the parabolic subgroup corresponding to the three leaves in the Coxeter–Dynkin diagram for $D_4$).

Classify the basic primitive groups $G$ which do not have the road closure property. In particular, is it true that such a group either has a subgroup of index at most 3 or is almost simple?