Nonabelian Brunn–Minkowski conjecture for simply connected simple Lie groups

Let $G$ be a simply connected simple Lie group equipped with a Haar measure $\mu$. Let $d$ be the dimension of $G$, and let $m$ be the dimension of a maximal compact subgroup of $G$.

Nonabelian Brunn–Minkowski conjecture. For every pair of compact sets $A,B\subseteq G$,

$$\mu(AB)^{\frac{1}{d-m}}\geq\mu(A)^{\frac{1}{d-m}}+\mu(B)^{\frac{1}{d-m}}.$$

This is the conjectural sharp Brunn–Minkowski inequality underlying the paper’s reduction arguments. The source explicitly says that the conjecture remains open in general, while a weaker inequality with a loss in the exponent is known.