Starlike rayless-tree characterization conjecture for compact generated ultrametric spaces

Let $(X,d)$ be an infinite ultrametric space. A labeled ray is a ray equipped with a labeling that generates the indicated ultrametric space. A tree is starlike if it has exactly one vertex of degree greater than $2$, and rayless if it contains no ray. Starlike rayless-tree characterization conjecture. The following statements are equivalent:

1. $(X,d)$ is the completion of a totally bounded proper subset $X_0\subsetneq X$ generated by a labeled ray.
2. There exists a starlike rayless tree $T$ with a labeling $l:V(T)\to\mathbb R^+$ such that $(X,d)$ is a compact ultrametric space generated by $T(l)$.

The claim is presented as a partial generalization of an earlier theorem in the source. No resolution status is supplied.