Highly generically transitive module conjecture

Let $(G,V)$ be a faithful module of finite Morley rank, and suppose that $V$ has Morley rank $n$ and $G$ is generically $n$-transitive. Highly generically transitive module conjecture. The module $(G,V)$ is definably equivalent to $(\operatorname{GL}_n(\mathbb{K}),\mathbb{K}^n)$ for some definable field $\mathbb{K}$. The conjecture is established for small values of $n$ and under sharpness assumptions, but remains open in general.