Metric characterization of ultrametric spaces generated by labeled trees

Let $(X,d)$ be a discrete, nonempty, totally bounded ultrametric space. Write $\mathbf{B}_X$ for the family of balls in $X$. A labeled tree is a tree $T=T(l)$ with vertex set $V(T)$ and labeling $l$, inducing the ultrametric $d_l$ on $V(T)$. The sphere centered at $c$ of radius $r$ is $S(c,r)=\{x\in X:d(x,c)=r\}$.

Metric characterization conjecture. The following statements are equivalent:

1. There is a labeled tree $T=T(l)$ such that $(V(T),d_l)$ and $(X,d)$ are isometric.
2. For every $B\in\mathbf{B}_X$, there are $c\in X$ and $r>0$ such that

$$B=\{x\in X:d(x,c)=r\}\cup\{c\}=S(c,r)\cup\{c\}.$$

The conjecture proposes a purely metric characterization of the discrete totally bounded ultrametric spaces representable by labeled trees. Its resolution status is not specified in the source.