Uniqueness of minimal constructible-closed boundaries for Tate Huber pairs

Determine whether every Tate Huber pair (A, A^{+}) admits a unique minimal constructible-closed boundary in its adic spectrum (relative to any choice of pseudo-uniformizer), i.e., prove or refute that the minimal constructible-closed boundary for (A, A^{+}) is unique and independent of auxiliary choices.

Background

The paper introduces a notion of boundary and Shilov boundary for Tate Huber pairs and proves existence of minimal constructible-closed boundaries (Proposition "Minimal closed boundaries"). In classical nonarchimedean analysis, uniqueness of minimal closed boundaries can be established under certain settings, but the authors extend the framework to general Tate Huber pairs where such arguments may not apply.

The authors note that while existence is ensured, it is unclear whether the minimal constructible-closed boundary is unique in this more general setting, highlighting a gap between classical results and the generalized context of adic spectra for Tate Huber pairs.