Completions are isometric iff positive-radius balleans are isometric

Determine whether, for ultrametric spaces (X, d) and (Y, \rho), the completions (\tilde{X}, \tilde{d}) and (\tilde{Y}, \tilde{\rho}) are isometric if and only if the balleans of closed balls with strictly positive radii (\bar{B}_X^{0}, d_H) and (\bar{B}_Y^{0}, \rho_H) are isometric.

Background

The set \bar{B}_X^{0} consists of all closed balls in (X, d) with strictly positive radii; in ultrametric spaces this set coincides with the isolated points of the ballean (Theorem 3.11) and forms a unique dense discrete subset (Proposition 3.21).

The conjecture proposes a characterization of the completion of an ultrametric space via the isometry class of the subspace (\bar{B}_X^{0}, d_H), suggesting that the geometry of positive-radius balls encodes all information needed to recover the completed space up to isometry.