Let $X$ be a locally compact Alexandrov space with curvature bounded below. Suppose $C$ is a closed subset that consists of the essential singular points, where a point $p$ is called an essential singular point if $\Sigma_p$ satisfies $\min_{\xi \in\Sigma_p } \max_{\eta \in \Sigma_p} \angle( \xi, \eta) \le \pi/2$.

For any $p\in C$ and $\xi \in \Sigma_p C$, can we prove that $\xi$ is an essential singular point of $\Sigma_p$?