Griesmer's inverse conjecture for Kneser's inequality in compact groups

Let $G$ be a compact group (not necessarily abelian) with probability Haar measure $\mu_G$, let $\varepsilon > 0$, and let $\delta>0$ be sufficiently small depending on $\varepsilon$. For any compact subsets $A,B \subset G$ satisfying

$$\mu_G(AB) \leq \mu_G(A)+\mu_G(B)+\delta$$

and

$$\mu_G(A) + \mu_G(B) \leq 1-\varepsilon,$$

there exist compact subsets $A',B' \subset G$ such that

$$\mu_G(A \mathbin{\Delta} A'),\\,\mu_G(B \mathbin{\Delta} B') \leq \varepsilon$$

and

$$\mu_G(A'B') \leq \mu_G(A') + \mu_G(B').$$

Griesmer's conjecture. Under these hypotheses, such compact subsets $A'$ and $B'$ exist. This conjecture proposes an inverse theorem for near equality in Kneser's inequality that extends the preceding abelian-group results to arbitrary compact, possibly nonabelian, groups. Its resolution is not given in the source.