Uniformity of Banach tangent cones in S-concave, locally semi-convex Busemann concave spaces

Determine whether, under the assumptions that X is S-concave, locally semi-convex, and Busemann concave with finite Hausdorff dimension n, all Banach tangent cones T_xX (at almost every or all points where they exist) are mutually isometric to the same finite-dimensional Banach space.

Background

The authors prove that in their setting X is n-rectifiable and H^n-almost every point admits a unique tangent cone isometric to a finite-dimensional Banach space. They then raise the question of whether these Banach tangent cones share a single isometry type across points.

Uniformity of tangent cone models would mirror rigidity phenomena in Finsler and Alexandrov contexts, yielding stronger global structural conclusions about the metric’s infinitesimal geometry across the space.