Borovik–Cherlin projective linearity conjecture for generically multiply transitive actions

Let $G$ be a connected group of finite Morley rank acting faithfully, definably, transitively, and generically $(n+2)$-transitively on a set $\Omega$ of Morley rank $n$.

Borovik–Cherlin's projective linearity conjecture. The pair $(G,\Omega)$ is equivalent to

$$\operatorname{PGL}_{n+1}(F)$$

acting on the projective space $\mathbb{P}^n(F)$ for some algebraically closed field $F$.

This is described as an almost inevitable next step after the paper's results. It remains open in the supplied source context.