Generic transitivity conjecture for primitive groups of finite Morley rank

Let $(G,X)$ be a connected primitive permutation group of finite Morley rank. Generic transitivity conjecture. Almost all such groups have maximum degree of generic transitivity at most $4$; the only exceptions are the groups listed in the rank bounds and classification conjecture. This is presented as a consequence suggested by comparison with finite primitive groups and Popov's theorem, but the meaning of “almost all” is not formally specified in the source.