Measure doubling of small sets in $\mathrm{S}\mathrm{O}(3,\mathbb{R})$

Let $\mathrm{S}\mathrm{O}(3,\mathbb{R})$ be the 3D-rotation group equipped with the real-manifold topology and the normalized Haar measure $\mu$. Resolving a problem by Breuillard and Green, we show that if $A\subseteq \mathrm{S}\mathrm{O}(3,\mathbb{R})$ is an open subset with sufficiently small measure, then $$\mu (A^2)>3.99\mu (A)\mathrm{.}$$ We also show a more general result for the product of two sets, which can be seen as a Brunn-Minkowski-type inequality for sets with small measure in $\mathrm{S}\mathrm{O}(3,\mathbb{R})$.