Strong Breuillard–Green conjecture for $\mathrm{SO}(3,\mathbb{R})$

Let $A\subseteq \mathrm{SO}(3,\mathbb{R})$ be open, and let $\mu$ denote normalized Haar measure on $\mathrm{SO}(3,\mathbb{R})$. For a unit vector $u\in\mathbb{R}^3$, define the angular displacement of $g$ at $u$ by $\angle(u,gu)$. Strong Breuillard–Green conjecture. One has

$$\mu(A^2)\geq \min\{1,4\mu(A)(1-\mu(A))\}.$$

Moreover, if $\mu(A)<1/2$, equality holds if and only if there is a unit vector $u\in\mathbb{R}^3$ such that

$$A=\{g\in \mathrm{SO}(3,\mathbb{R}):\angle(u,gu)<\arccos(1-\mu(A))\}.$$

This strengthens the small-set Breuillard–Green phenomenon by proposing the optimal doubling bound for every open set in $\mathrm{SO}(3,\mathbb{R})$, together with a characterization of the equality cases for sets of measure below $1/2$.