Cauchy subsequences and hulls of sequences in labeled trees

Let $T$ be a tree, and let $(v_n)_{n\in\mathbb{N}}$ be a sequence of distinct vertices of $T$. For a set of vertices, its hull is the corresponding minimal connected subgraph. A labeling $l:V(T)\to\mathbb{R}$ is non-degenerate in the sense used for the ultrametric induced by $l$.

Cauchy-hull conjecture. The following conditions are equivalent:

1. The hull of the range set of $(v_n)_{n\in\mathbb{N}}$ is a union of a ray with some finite tree.
2. For every non-degenerate $l:V(T)\to\mathbb{R}$, the existence of a Cauchy subsequence of $(v_n)_{n\in\mathbb{N}}$ implies that $(v_n)_{n\in\mathbb{N}}$ is itself a Cauchy sequence.

This conjecture links the geometry of the hull of a sequence's range to the behavior of Cauchy subsequences in the ultrametrics induced by labelings. Its resolution status is not specified in the source.