Uniformization conjecture for strongly asymptotically log del Pezzo pairs

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

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

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

i.e. if and only if $(S,C)$ is of class $(\aleph)$ or $(\beth)$.

This conjecture proposes that the positivity classification controls the small-angle existence theory. The preceding alpha-invariant theorem supports it by giving limiting values $1$, $1/2$, and $0$ for the four positivity classes, while the existence assertion itself remains open in the source.