The $K_{1,3}$ characterization conjecture for ultrametric spaces

Let $(X,d)$ be an ultrametric space with $|X|\geq 4$. For each four-point subspace, let its diametrical graph be the graph formed from the four points using the paper's definition. Let $(Z_4,\mu_4)$ and $(S_4,\nu_4)$ be the four-point ultrametric spaces displayed in the paper, and let “weakly similar” have the paper's stated meaning.

$K_{1,3}$ characterization conjecture. The following statements are equivalent:

1. The diametrical graph of each four-point subspace of $(X,d)$ is isomorphic to $K_{1,3}$, but $(X,d)$ contains no four-point subspaces which are weakly similar to $(Z_4,\mu_4)$.
2. All four-point subspace of $(X,d)$ are weakly similar to $(S_4,\nu_4)$.
3. The space $(X,d)$ admits an isometric embedding into $(\mathbb R^+,d^+)$.

This conjecture seeks to characterize the ultrametric spaces whose four-point patterns force an isometric embedding into the positive real line. The source calls it a plausible hypothesis, and no resolution is supplied in the given text.