The continuous Babai conjecture for compact simple groups

Let $G$ be a compact simple group, let $r$ be the dimension of one of its maximal tori, and let $A\subset G$ be a measurable set.

Continuous Babai conjecture. There is an integer $m$ such that

$$A^m=G,$$

with

$$m=O\!\left(\frac{\log\mu_G(A)}{r_G}\right),$$

where the implied constants do not depend on $G$.

This is presented as a continuous analogue of Babai's conjecture for finite simple groups in the unbounded-rank setting. The source does not provide evidence of a resolution for this formulation.