Stability of magnitude for finite subsets of Euclidean and taxicab spaces

Prove that for U = (ℝ^n, ℓ^p) with 1 ≤ p ≤ 2, the magnitude mapping X ↦ |X| is continuous with respect to the Hausdorff topology on the set of finite subsets of U, thereby establishing stability of magnitude for finite subsets of Euclidean (ℓ^2) and taxicab (ℓ^1) spaces.

Background

The conjecture isolates a practically important setting: finite subsets in ℝ^n with ℓ^p metrics (p ∈ [1,2]), which are positive definite and central to applications in data analysis and machine learning. A proof would furnish a robust stability theorem compatible with standard embedding contexts.

This conjecture is motivated by both the failure of generic continuity in Formulation A and the partial success along lines in Formulation B, suggesting that continuity may hold in widely used ambient spaces despite pathological behavior in general Gromov–Hausdorff space.