  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
$\operatorname{PGL}_{n+1}(\mathbb{F})$ 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
$(\mathbb{P}^{n-1}(\mathbb{F}),\operatorname{PGL}_{n}(\mathbb{F}))$. The latter
hypothesis drives the construction of the underlying projective geometry and is
at the heart of an inductive approach to the main problem.
