Generic transitivity degree is ≤ 4 for almost all primitive actions

Show that, except for the actions listed in Conjecture 7, almost all connected primitive permutation groups of finite Morley rank have maximum degree of generic transitivity at most 4.

Background

This conjecture parallels Maróti’s order bounds for finite primitive groups and leverages Popov’s classification of maximal generic transitivity for simple algebraic groups in characteristic 0.

It posits that high generic transitivity is rare in the finite Morley rank setting, concentrating in a small family of algebraic-type actions.