Approximate subgroup containment in a subgroup neighbourhood

Determine whether there exist constants C > 0 (depending only on c > 0) such that for every compact Lie group G, if A ⊂ G is a measurable exp(c(d_G−d_H))-approximate subgroup of measure at most exp(−C(d_G−d_H)), then A is contained in H^2 for some proper closed subgroup H of G.

Background

This problem seeks a structural criterion for very small approximate subgroups in compact Lie groups, with quantitative dependence on d_G−d_H (the codimension of a maximal proper subgroup). The authors explain that such a result would be optimal up to sharp examples and would imply the continuous Babai-type growth conjecture via known arguments.