Cheltsov–Rubinstein conjecture on small-angle Kähler–Einstein edge metrics

Let $X$ be a smooth projective variety and let $D$ be a smooth divisor such that $(X,D)$ is asymptotically log Fano. For $\beta\in(0,1]$, write $(X,(1-\beta)D)$ for the corresponding log pair; a Kähler–Einstein edge metric is a metric with cone angle $2\beta a\pi$ along $D$. Cheltsov–Rubinstein conjecture. The pair $(X,(1-\beta)D)$ admits a Kähler–Einstein edge metric with sufficiently small cone angle $\beta$ along $D$ if and only if

$$(K_X+D)^{{\operatorname{dim}} X}=0.$$

The conjecture concerns the existence of canonical metrics in the small-cone-angle limit for asymptotically log Fano pairs. The source states that it does not hold in general and that counterexamples are presented, so it is refuted.