  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
$\mathcal C=W$ 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 $\mathcal C=W$ in the two-step
stable degeneration of $(X,p)$ and $\mathcal C$ has a smooth link, then every
singular K\"ahler-Einstein metric on $X$ with non-positive Ricci curvature and
bounded potential is conical at $p$.
