A classification of finite partial linear spaces with a primitive rank 3 automorphism group of grid type

A group of grid type is a permutation group $G$ acting on $\Omega \times \Omega$ ($\Omega$ is a finite set of size $n$) such that $S\times S⊲G⩽S_0\wr 2$, where $S_0$ is 2-transitive on $\Omega$ and almost simple with $S⊴S_0\le \mathrm{Aut}(S)$.

We can visualize $G$ as acting on the $n\times n$ grid. Note that there is no 2-transitive almost simple group of degree 4 or less, so we may assume $n\ge 5$. The finite $2$-transitive almost simple groups have been classified in List of Almost Simple 2-transitive Group. In every case (except for $S_0=P\Gamma L(2,8)$ in its action on 28 points), the simple group $S$ is $2$-transitive.