Zero-angle limit conjecture for Kähler–Einstein pairs on log del Pezzo surfaces

Let $(S,C)$ be a log del Pezzo surface pair in class $(\aleph)$ or $(\beth)$, and let $\omega_\beta$ be a conical Kähler–Einstein metric with cone angle $2\pi\beta$ along $C$. In class $(\aleph)$, $S$ is del Pezzo with $C\sim-K_S$; in class $(\beth)$, $C\not\sim-K_S$ and $(K_S+C)^2=0$.

Zero-angle limit 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.

The conjecture generalizes the folklore expectation that the Tian–Yau metric on $S\setminus C$ is a limit of conical Kähler–Einstein metrics when $(S,C)$ is in class $(\aleph)$. The supplied text gives no resolution status.