The bounded generic-base-gap conjecture for finite Morley rank actions

Let $(G,S)$ be a definably primitive permutation group of finite Morley rank. Let $b(G,S)$ be the minimal cardinality of a base, and let $b^1(G,S)$ be the least positive integer $n$ such that $S^n$ contains a generic subset all of whose tuples are bases. Bounded generic-base-gap conjecture. There is a natural number $d$ such that

$$b^1(G,S)-b(G,S)<d.$$

This proposed amendment allows the generic base size to exceed the minimal base size, as happens in many algebraic examples, while conjecturing that the gap is uniformly bounded. The source gives no resolution of this formulation.