Isometry of space and its ballean implies equidistance?

Determine whether the metric d on an ultrametric space (X, d) must be equidistant whenever the ballean (\bar{B}_X, d_H) and the original space (X, d) are isometric.

Background

Theorem 4.7 shows that when d is equidistant (all pairwise distances between distinct points are equal), the spaces (X, d) and (\bar{B}_X, d_H) are isometric, except for a finite nontrivial case. This question asks for the converse implication: does isometry of the space and its ballean force d to be equidistant?

An affirmative answer would classify all ultrametric spaces whose balleans are isometric to themselves, while a negative answer would require counterexamples and further structural analysis.