Borovik–Cherlin conjecture on generically transitive actions on connected abelian groups

Let $G$ be a connected group of finite Morley rank acting faithfully, definably, and generically $n$-transitively on a connected abelian group $V$ of Morley rank $n$.

Borovik–Cherlin's conjecture. The group $V$ has the structure of an $n$-dimensional vector space over an algebraically closed field $F$ of Morley rank $1$, and

$$G=\operatorname{GL}_n(F)$$

with its natural action on $F^n$.

The theorem proved in the paper gives a partial confirmation of this conjecture. The source notes that Altınel and Wiscons have already made important contributions toward its solution, but the conjecture is not presented as fully resolved here.