The Siemons–Wagner conjecture on primitive permutation groups and subset orbit sizes

Let $G$ be a primitive permutation group acting on a set $\Omega$ of cardinality $n\ge 8$. For a subset $S\subseteq \Omega$, write $S^G$ for its orbit under $G$. Siemons–Wagner conjecture. If there exists a $3$-subset $\Delta\subseteq \Omega$ such that

$$|\Delta^G|>|\Sigma^G|$$

for every $4$-subset $\Sigma$ containing $\Delta$, then $G\cong PSL(2,7)$ or $G\cong PGL(2,7)$. This conjecture identifies the two exceptional primitive groups whose orbit structure on $3$- and $4$-subsets has the stated inequality; the surrounding argument establishes the claim for the remaining families considered there, while the full assertion is left as a conjecture.