Donaldson–Sun canonicity conjecture for metric tangent cones

Let $M_\infty$ be a Gromov–Hausdorff limit of Kähler–Einstein Fano manifolds, let $o\in M_\infty$, and let $C=C_oM_\infty$ be a metric tangent cone at $o$. Let $R=\mathcal{O}_{M_\infty,o}$, and let $W$ be the affine variety obtained from the finitely generated associated graded ring of the filtration of $R$ defined by the limiting metric; Donaldson–Sun show that $W$ equivariantly degenerates to $C$. Donaldson–Sun's conjecture. Both $W$ and $C$ depend only on the algebraic structure of $M_\infty$ near $o$. This concerns the canonicity of the intermediate algebraic cone and the metric tangent cone. Their construction uses the limiting metric, and the conjecture asserts that the resulting objects are nevertheless determined by the local algebraic germ.