The generic-type conjecture for simple superstable groups

Let $G$ be a simple superstable group, and suppose its $U$-rank is of the form

$$U(G)=\omega^{\alpha+1}\cdot n.$$

Here a generic type of $G$ is a complete type describing a generic element of the group, and ESN is the property used in the source.

Generic-type conjecture. The generic type of $G$ is ESN.

The conjecture is presented as a proposed affirmative answer to a question about generic types of simple superstable groups. The supplied text gives no resolution status and does not define ESN in the excerpt.