Cheltsov–Rubinstein's Kähler–Einstein edge metric conjecture for strongly asymptotically log del Pezzo surfaces

Let $(S,C)$ be a strongly asymptotically log del Pezzo surface with $r=1$, where $S$ is a smooth surface and $C$ is a smooth irreducible curve. The notation $(K_S+C)^2$ denotes the self-intersection number of the log canonical divisor $K_S+C$. Cheltsov–Rubinstein's conjecture. The pair $(S,C)$ admits a Kähler–Einstein edge metric for sufficiently small $\beta$ if and only if

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

This conjecture was proposed as a step towards the classification problem for strongly asymptotically log del Pezzo surfaces. Its resolution is not specified in the source.