Rooted-tree characterization of finite homogeneous ultrametric spaces

Prove or disprove that a finite rooted tree represents a finite homogeneous ultrametric space if and only if no vertex has out-degree one and any two vertices at the same level have the same out-degree.

Background

Finite ultrametric spaces can be represented by labeled rooted trees, while the final conjecture concerns the underlying rooted-tree structure of finite homogeneous ultrametric spaces. The proposed condition requires that every vertex have out-degree different from one and that out-degree be constant on each level.

The conjecture asks for an equivalence between this purely rooted-tree condition and representability by a finite homogeneous ultrametric space. It remains unresolved in the paper.