Closed-ball characterization via distance sets in ultrametric spaces

Let $(X,d)$) be an ultrametric space. For each $p\to X$, define

$$D_p(X):=\{d(p,x):x\in X\}.$$

Let $\overline{\bf B}_X$ denote the set of all closed balls of $(X,d)$, and let $\bf Cs_X$ denote the family of centered spheres of $(X,d)$. The closed-ball characterization conjecture. If

$$\overline{\bf B}_X\subseteq\bf Cs_X,$$

then $D_p(X)\cap[0,r]$ has a greatest element for every $p\in X$ and $r\in\mathbb R^+$. Conversely, if $(X,d)\in\bf UT$ and $D_p(X)\cap[0,r]$ has a greatest element for every $p\in X$ and $r\in\mathbb R^+$, then

$$\overline{\bf B}_X\subseteq\bf Cs_X.$$

This conjecture seeks an intrinsic distance-set criterion for all closed balls to be centered spheres; the source gives no resolution status.