Doubling minimiser among sets of fixed measure

Determine whether, for every compact Lie group G, every proper closed subgroup H, and every δ > 0, the neighbourhood H_δ minimises the doubling ratio µ_G(A^2)/µ_G(A) among all measurable subsets A ⊂ G with µ_G(A) = µ_G(H_δ).

Background

The paper proves sharp lower bounds and stability for minimal doubling of small subsets in compact Lie groups, showing that neighbourhoods of maximal subgroups achieve close-to-optimal doubling and characterising near-minimisers. Motivated by these results, the authors conjecture that the optimal (exact) minimisers of the doubling ratio at fixed measure are precisely neighbourhoods of subgroups.