Three-point characterization of ultrametric spaces whose subsets are centered spheres

Let $(Y,\rho)$ be an ultrametric space with $|Y|\geq3$. A centered sphere is a sphere with a center in the underlying space. The three-point centered-sphere conjecture. The following are equivalent: every nonempty subset of $Y$ is a centered sphere in $(Y,\rho)$; and $Y$ has exactly three points and $(Y,\rho)$ is weakly similar to the specified three-point ultrametric space $(X_3,d)$. The source presents this as a characterization, but gives no resolution status.