The generic pair conjecture for dependent theories

Assume that $T$ is a first-order theory and that

$$\lambda=\lambda^{<\lambda}>|T|,$$

with $2^\lambda=\lambda^+$. Let $\langle M_\alpha:\alpha<\lambda^+\rangle$ be a $\prec$-increasing continuous sequence with $M_\alpha\in\operatorname{EC}_\lambda(T)$, and suppose that

$$\bigcup_{\alpha<\lambda^+}M_\alpha\in\operatorname{EC}_{\lambda^+}(T)$$

is saturated. Generic pair conjecture. The theory $T$ is dependent if and only if there is a club $E\subseteq\lambda^+$ such that for all $\alpha<\beta$ in $E$ of cofinality $\lambda^+$, the pairs $(M_\beta,M_\alpha)$ have the same isomorphism type. The conjecture is presented as the structure-side counterpart to non-structure results for independent theories. The source states that the structure side is proved when $\lambda=\kappa$ is measurable, while the general conjecture remains open.