Classification conjecture for simple groups with a generically 4-transitive action

Let $G$ be a simple group of finite Morley rank admitting a generically $4$-transitive action. Generically 4-transitive classification conjecture. Either $G\simeq\operatorname{PGL}_3$ or $G\simeq E_6$. This is a proposed high-transitivity special case of the broader classification programme; the survey presents it as open.