The compactness characterization conjecture for infinite labeled star ultrametric spaces

Let $(X,d)$ be an infinite ${\bf US}$-space generated by a labeled star graph with center $c$. Set

$$X_0:=X\setminus\{c\},$$

and let $d_0$ be the restriction of $d$ to $X_0\times X_0$. A labeled ray $R(l)$ has edge set $E(R)$ and labeling function $l$.

Compactness characterization conjecture. The following statements are equivalent: $(X,d)$ is compact; and there is a labeled ray $R(l)$ generating $(X_0,d_0)$ such that

$$E(R)=\{\{x_1,x_2\},\{x_2,x_3\},\dots,\{x_n,x_{n+1}\},\dots\}$$

with

$$l(x_n)\geq l(x_{n+1})>0$$

for every integer $n\in\mathbb N$, and

$$\lim\limits_{n\to\infty}l(x_n)=0.$$

This would characterize compactness of an infinite ultrametric space in ${\bf US}$ through the ray structure and decay of its labels. The source presents the equivalence as a conjecture and supplies no evidence of a resolution.