Asymptotic logarithmic Kähler–Einstein conjecture for log del Pezzo surfaces

Let $(S,C)$ be a strongly asymptotically log del Pezzo pair with $C$ smooth and irreducible. A conical Kähler–Einstein metric on $(S,C)$ is a Kähler–Einstein metric with cone angle $2\pi\beta$ along $C$.

Asymptotic logarithmic Kähler–Einstein conjecture. The surface $S$ admits conical Kähler–Einstein metrics with angle $\beta$ along $C$ for all sufficiently small $\beta$ if and only if

$$(K_S+C)^2=0.$$

This is the proposed counterpart of Tian's classification result for smooth del Pezzo surfaces, in the strongly asymptotically log del Pezzo setting. The supplied text gives no resolution status.