Asymptotic curvature-set similarity

Determine whether the Hausdorff distance between the curvature sets K_{n+2}(B^{n+1}) and K_{n+2}(B^n) converges to zero as n tends to infinity.

Background

Curvature sets encode finite distance matrices realized by a metric space and provide lower bounds for Gromov–Hausdorff distance. The paper leaves open whether the specific curvature sets of consecutive-dimensional Euclidean balls become asymptotically indistinguishable in Hausdorff distance as the dimension grows.

References

Does $d_H(K_{n+2}(B{n+1}),K_{n+2}(Bn))\rightarrow0$ as $n\rightarrow\infty$?

— Gromov--Hausdorff Distance Between Euclidean Unit Balls  (2609.09652 - Adams et al., 9 Sep 2026) in Question 2, Section 8 (Using curvature sets of balls to lower bound d_GH)