Finite four-point obstruction conjecture for labeled star graph ultrametric spaces

Let $(X^*,d^*)$ be a finite ultrametric space. The space $X^*$ is a US-space when it belongs to the class ${\bf US}$ of ultrametric spaces generated by labeled star graphs. A four-point subspace is weakly similar to another metric space when there is a similarity between them, allowing a positive rescaling of the distance. Finite four-point obstruction conjecture. The following statements are equivalent:

1. $(X^*,d^*) \notin {\bf US}$.
2. $(X^*,d^*)$ contains a four-point subspace which is weakly similar either to $(X_4,d_4)$ or to $(Y_4,\rho_4)$.

This conjecture proposes that the two displayed four-point spaces are the complete finite obstructions to membership in ${\bf US}$; its resolution is not supplied in the source.