Uniqueness of geodesics in the Alexandrov space

Question

Let $X$ be an Alexandrov space with curvature bounded below. For any point $p \in X$, we define $C_p$ to be the set that consists of all points $q$ such that there are at least two minimizing geodesics from $p$ to $q$.

Can we prove that for any $p \in X$ there exists an open neighborhood $U$ of $p$ such that $U \bigcap C_p=\emptyset$?

Answer 1

I believe the example of Otsu-Shioya in page 632 of https://projecteuclid.org/download/pdf_1/euclid.jdg/1214455075 shows $C_p$ could be dense in the space.