Recognition conjecture for generically highly transitive groups of finite Morley rank

Let $(X,G)$ be a permutation group of finite Morley rank, where $G$ acts transitively on $X$. The action is generically $n$-transitive if $G$ has an orbit on $X^n$ whose complement has rank strictly less than $\operatorname{rk}(X^n)$. Assume that $G$ is connected and that

$$\operatorname{rk} X=n.$$

Recognition conjecture. If $(X,G)$ is transitive and generically $(n+2)$-transitive, then $(X,G)$ is equivalent to

$$(\operatorname{\mathbb{P}}^{n}(K),\operatorname{PGL}_{n+1}(K))$$

for some algebraically closed field $K$. This conjecture proposes a natural upper limit for generic transitivity in the finite Morley rank setting and characterizes the projective linear action as the extremal case. Its status is not determined by the supplied source material.