The generic pair conjecture for dependent diagrams

Fix a dependent diagram $D$ and a strongly compact cardinal $\theta>|T|$. Let $\theta<\lambda=\lambda^{<\lambda}$ and $\lambda^{+}=2^{\lambda}$. Let

$$\bar{M}=\langle M_{\alpha}:\alpha<\lambda^{+}\rangle$$

be an increasing continuous sequence of elementary substructures of $\mathfrak{C}_{D}$, each of cardinality $\lambda$, such that

$${\bf M}=\bigcup_{\alpha<\lambda^{+}}M_{\alpha}$$

is $D$-saturated of size $\lambda^{+}$. The generic pair conjecture. There exists a club $E\subseteq\lambda^{+}$ such that whenever $\alpha_{1}<\beta_{1}$ and $\alpha_{2}<\beta_{2}$ are in $E$ and all four ordinals have cofinality $\lambda$, one has

$$(M_{\beta_{1}},M_{\alpha_{1}})\cong(M_{\beta_{2}},M_{\alpha_{2}}).$$

This is the finite-diagram generalization of the generic pair conjecture; the paper studies it for dependent diagrams using homogeneous models and strong compactness. Its status is not resolved in the supplied text.