The conjecture on tangent cones and metric density of log Kähler–Einstein limits

Let $(W,(1-\beta)\Delta,g)$ be a Gromov–Hausdorff limit of log Fano pairs admitting a Kähler–Einstein metric. For a point $p$ of this metric space, let the metric tangent cone at $p$ and its metric density be denoted by the corresponding tangent-cone notation and $\Theta_p$, and let $\widehat{\operatorname{vol}}_{W,p}$ denote the normalised volume of the singularity.

Tangent-cone conjecture. The following properties hold:

1. For every point $p\in (W,(1-\beta)\Delta,g)$, the metric tangent cone at $p$ is unique.
2. The metric density satisfies

$$\Theta_p=\widehat{\operatorname{vol}}_{W,p}.$$

3. The metric tangent cone admits an algebraic interpretation via the two-step construction.

These are proposed as log-setting analogues of results known in the absolute case. The source says that the picture is incomplete, in particular because the corresponding tangent-cone results for log pairs were not yet proved.