Classification conjecture for maximally generically transitive actions

Let $G$ be a connected group of finite Morley rank acting faithfully, definably and transitively on a set $X$ of Morley rank $n$, and suppose the action is generically $(n+2)$-transitive. Maximal generic-transitivity conjecture. The permutation group $(G,X)$ is definably equivalent to $(\operatorname{PGL}_{n+1}(\mathbb{K}),\mathbb{P}^n(\mathbb{K}))$ for some definable field $\mathbb{K}$. This conjecture explains the expected sharp bound on generic transitivity. It is known for $n=1$ and $n=2$, while the general case remains open.