Small-angle metric degeneration conjecture for classes $(\aleph)$ and $(\beth)$

Let $(S,C,\omega_\beta)$ be Kähler–Einstein edge pairs of class $(\aleph)$ or $(\beth)$, where $\omega_\beta$ has angle $\beta$ along $C$. Let $\omega_\infty$ denote a generalized Kähler–Einstein metric on the complement $S\setminus C$.

Small-angle metric degeneration conjecture. As $\beta$ tends to zero, $(S,C,\omega_\beta)$ converges in an appropriate sense to a generalized Kähler–Einstein metric $\omega_\infty$ on $S\setminus C$. In class $(\aleph)$, $\omega_\infty$ is a Calabi–Yau metric; in class $(\beth)$, it is a cylinder along each generic fiber.

This conjecture describes the expected geometric limit underlying the uniformization conjecture. The source does not specify the precise notion of convergence, and the assertion remains open there.