The minimiser conjecture for doubling in compact Lie groups

Let $G$ be a connected Lie group, let $H$ be a proper closed subgroup of maximal dimension, and for $\delta>0$ define its $\delta$-neighbourhood by

$$H_{\delta}:=\{g\in G:d(g,H)<\delta\}.$$

For a measurable subset $A\subset G$, its doubling constant is $\mu_G(A^2)/\mu_G(A)$.

Doubling minimiser conjecture. The subset $H_{\delta}$ minimises the doubling constant among measurable subsets $A$ satisfying

$$\mu_G(A)=\mu_G(H_{\delta}).$$

The conjecture concerns the precise minimisers underlying the sharp small-set doubling bounds. The source presents it as a conjecture and does not state a resolution.