Completion of ballean equals ballean of completion

Establish whether, for every ultrametric space (X, d), the ballean (\bar{B}_{\tilde{X}}, \tilde{d}_H) of the completion (\tilde{X}, \tilde{d}) is isometric to the completion of the ballean (\bar{B}_X, d_H), where (\bar{B}_X, d_H) denotes the metric space of all closed balls of (X, d) endowed with the Hausdorff metric.

Background

The paper proves that an ultrametric space (X, d) is complete if and only if its ballean (\bar{B}_X, d_H) is complete (Theorem 3.13). Motivated by this parallel, the conjecture asks whether completing the space before forming its ballean yields the same (up to isometry) result as forming the ballean first and then completing it.

Here the ballean refers to the space of all closed balls of (X, d) equipped with the Hausdorff metric, and (\tilde{X}, \tilde{d}) is the metric completion of (X, d). The conjecture seeks a canonical identification between these two constructions.