Embeddability into US via four-point obstructions

Prove the equivalence for every infinite ultrametric space (X, d) that: (i) there exists an ultrametric space (X*, d*) generated by a labeled star graph (i.e., a member of the class US of ultrametric spaces generated by labeled star graphs) such that (X, d) embeds isometrically into (X*, d*); and (ii) (X, d) contains no four-point subspace that is weakly similar (via a strictly increasing bijection on distance sets) to either the four-point ultrametric space (X4, d4) or the four-point ultrametric space (Y4, P4).

Background

The class US consists of ultrametric spaces generated by labeled star graphs, characterized by the four-point condition in Theorem 2.14. The paper proves a precise four-point characterization for finite spaces and compact spaces (Theorems 4.3 and 5.2), showing that for such spaces, absence of four-point subspaces weakly similar to (X4, d4) or (Y4, P4) is equivalent to belonging to US.

Conjecture 6.1 extends this obstruction-based characterization to all infinite ultrametric spaces by asking for equivalence between the absence of these four-point obstructions and isometric embeddability into some US-space. The paper notes that the implication from embeddability to absence of obstructions follows from Proposition 4.2, and that the conjecture holds in the compact case by Theorem 5.2. The remaining challenge is the non-compact case.