Dimension-independent quantitative stability constant

Ascertain whether the quantitative stability bound in Theorem 1.4 can be made dimension-independent, i.e., obtain a dependency δ = C(c)α with the constant c independent of the ambient compact Lie group G.

Background

The paper proves qualitative stability for near-minimal doubling and sketches quantitative directions, noting current dependencies on the group dimension. This problem asks whether one can remove that dependence, aligning with sharper stability phenomena known in Euclidean settings.