On polynomial convergence to tangent cones for singular K\"ahler-Einstein metrics

Let (Z, p) be a pointed Gromov-Hausdorff limit of non-collapsing K\"ahler-Einstein metrics with uniformly bounded Ricci curvature. We show that the singular K\"ahler-Einstein metric on Z is conical at p if and only if C=W in Donaldson-Sun's two-step degeneration theory, assuming curvature grows at most quadratically near p. Let (X, p) be a germ of an isolated log terminal algebraic singularity. Following Hein-Sun's approach, we show that if C=W in the two-step stable degeneration of (X, p) and C has a smooth link, then every singular K\"ahler-Einstein metric on X with non-positive Ricci curvature and bounded potential is conical at p.