Characterization of homogeneous ultrametric spaces by self-isometry count

Determine whether, for a finite homogeneous ultrametric space \((X,d)\), the equality \(\log_2|\operatorname{Is}(X)|=|X|-1\) holds if and only if \((X,d)\) is an MCD-space.

Background

This conjecture seeks a converse to the proposed self-isometry formula for MCD-spaces. It restricts attention to finite homogeneous ultrametric spaces and asks whether the exact value of the self-isometry group characterizes the extremal MCD condition.

The problem is explicitly presented as requiring proof or disproof, so the claimed equivalence is unresolved in the paper.

References

Let (X,d) be a finite homogeneous ultrametric space. Then equality lli holds if and only if (X,d)\in {\bf MCD}.

— Maximal Center of Distances of Finite Ultrametric Spaces and Perfect Binary Trees  (2609.11384 - Dovgoshey et al., 10 Sep 2026) in Section 5, second Conjecture (labeled “Prove or disprove”)