Donaldson–Sun's algebraic determination conjecture for metric tangent cones

Let $(X,d_\infty)$ be a Gromov–Hausdorff limit as in the construction of Donaldson and Sun, let $x\in X$ be a point, let $C=C_xX$ be a metric tangent cone, and let $W$ be the affine variety obtained from the associated graded ring of the filtration of the local ring determined by the limiting metric. Donaldson–Sun's conjecture. Both $W$ and $C$ depend only on the algebraic germ structure of $X$ near $x$. This conjecture asks for the metric tangent-cone construction to be determined algebraically rather than by the limiting metric.