Completion-determination conjecture for the isolated closed balls of ultrametric spaces

Let $(X,d)$ and $(Y,\rho)$ be ultrametric spaces, with completions $(\tilde{X},\tilde{d})$ and $(\tilde{Y},\tilde{\rho})$. Let $\bar{\mathbf{B}}_X^0$ and $\bar{\mathbf{B}}_Y^0$ denote the subspaces of isolated points in the balleans of closed balls, equipped with the Hausdorff metrics $d_H$ and $\rho_H$, respectively. Completion-determination conjecture. The following statements are equivalent: 1. The completions $(\tilde{X},\tilde{d})$ and $(\tilde{Y},\tilde{\rho})$ are isometric. 2. The ultrametric spaces $(\bar{\mathbf{B}}_X^0,d_H)$ and $(\bar{\mathbf{B}}_Y^0,\rho_H)$ are isometric. This proposed characterization says that the isometry type of the completion is exactly captured by the isolated closed balls, but the paper provides no resolution.