Towards the recognition of ${\mathrm{PGL}}_n$ via a high degree of generic transitivity

In 2008, Borovik and Cherlin posed the problem of showing that the degree of generic transitivity of an infinite permutation group of finite Morley rank $(X,G)$ is at most $n+2$ where $n$ is the Morley rank of $X$. Moreover, they conjectured that the bound is only achieved (assuming transitivity) by ${\mathrm{PGL}}_{n+1}(\mathbb{F})$ acting naturally on projective $n$-space. We solve the problem under the two additional hypotheses that (1) $(X,G)$ is $2$-transitive, and (2) $(X-\{x\},G_x)$ has a definable quotient equivalent to $({\mathbb{P}}^{n-1}(\mathbb{F}),{\mathrm{PGL}}_n(\mathbb{F}))$. The latter hypothesis drives the construction of the underlying projective geometry and is at the heart of an inductive approach to the main problem.