When do ballean isometries come from underlying space isometries?

Identify conditions on isometric ultrametric spaces (X, d) and (Y, \rho) that ensure every isometry \mathbf{F}: \bar{B}_X \to \bar{B}_Y between their balleans is induced by an isometry F: X \to Y via \mathbf{F}(\bar{B}) = \{F(x) : x \in \bar{B}\} for all \bar{B} \in \bar{B}_X.

Background

If F: X \to Y is an isometry, then the induced map \bar{B}_X \ni \bar{B} \mapsto {F(x) : x \in \bar{B}} \in \bar{B}_Y is an isometry between balleans. The problem asks for the converse: under what hypotheses does every ballean isometry arise from an underlying space isometry in this canonical way?

Remark 6.5 notes that for equidistant spaces with |X| \neq 1, one can find ballean isometries that are not induced by any space isometry, indicating that additional structural conditions are necessary.