Is the following consistent?
CH fails and there is a partial order $\preceq$ on $\omega_2$ such that for every $\alpha < \omega_2$, $(\alpha, \preceq)$ can be extended to a separable linear order $(\alpha, \preceq_{\alpha})$ but there is no separable linear order extending $(\omega_2, \preceq)$.
This is related to an earlier question of Noah Schweber: What kind of compactness does "expanding $\mathbb{R}$ by constants" have?