The Unique Limit Model Conjecture for dependent theories

Let $T$ be dependent, let $|T|<\lambda=\lambda^{<\lambda}$, let $\lambda^+=2^\lambda$, and let $\sigma=\operatorname{cf}(\sigma)<\lambda$. Suppose $\langle M_\alpha:\alpha<\lambda^+\rangle$ is an increasing continuous sequence of models of cardinality $\lambda$ whose union is $\lambda^+$-saturated. The Unique Limit Model Conjecture. There is a club $E\subseteq\lambda^+$ such that all models $M_\alpha$ with $\alpha\in E$ and $\operatorname{cf}(\alpha)=\sigma$ are pairwise isomorphic. This predicts uniqueness, on a club of stages of fixed cofinality, for limit models arising in dependent theories; the source supplies no resolution in the displayed passage.