The Lascar-rank conjecture for countable models of superstable theories

Let $T$ be a complete, superstable theory in a countable language with infinite models. Write $I(T,\aleph_{0})$ for the number of countable models of $T$, and let $U(T)$ denote the supremum of the Lascar ranks of complete types of $T$.

Lascar-rank conjecture. If

$$U(T)\geq\omega^{\omega},$$

then

$$I(T,\aleph_{0})=2^{\aleph_{0}}.$$

This conjecture concerns the relationship between the Lascar rank of a superstable theory and the number of its countable models. The supplied text attributes it to an earlier work but gives no resolution status.