Continuous Babai-type growth in compact simple Lie groups

Establish that for every compact simple Lie group G and measurable set A ⊂ G, there exists m = O((log(1/µ_G(A)))/r) such that A^m = G, where r is the dimension of a maximal torus of G and the implied constants are independent of G.

Background

The conjecture is a continuous analogue of Babai's conjecture on growth in finite simple groups, formulated for compact simple Lie groups. The authors reference recent partial progress for large sets and present their own small-set growth bound (Proposition 8.5), noting that achieving a uniform exponential-scale bound on a key parameter would imply the conjecture.