The UGVL characterization conjecture for totally bounded ultrametric spaces

Let $(X,d)$ be a nonempty totally bounded ultrametric space. A point $x\in X$ is isolated if it has a neighborhood containing no point of $X$ other than $x$. Let $\mathbf{B}_{X,d}$ be the set of open balls of $(X,d)$, let $\mathbf{Cs}_{X,d}$ be the set of centered spheres, and let $\mathbf{UGVL}$ denote the class of ultrametric spaces generated by nonnegative vertex labelings of trees. Assume that every point of $X$ is isolated.

UGVL characterization conjecture. The following statements are equivalent:

1. $(X,d)\in\mathbf{UGVL}$.
2. $\mathbf{B}_{X,d}\subseteq\mathbf{Cs}_{X,d}$.

This characterizes, for nonempty totally bounded ultrametric spaces with isolated points, when the space is generated by a labeled tree in terms of its open balls being centered spheres. E. Petrov proved the conjecture for finite ultrametric spaces using a different terminology and the technique of representing trees due to Gurvich and Vyalyi.