Diameter conjecture for large subsets of classical compact groups

Let $G$ be one of $\mathrm{SU}(n)$, $\mathrm{SO}(n)$, $\mathrm{Spin}(n)$, or $\mathrm{Sp}(n)$, and let $\mathcal{A}\subseteq G$ be measurable with measure $\nu$. The diameter with respect to $\mathcal{A}$ is the smallest positive integer $t$ such that $\mathcal{A}^t=G$. Diameter conjecture. The diameter of $G$ with respect to $\mathcal{A}$ is $O(\nu^{-1/(\ell n)})$, where $\ell=1$ for $\mathrm{SO}(n)$ and $\mathrm{Spin}(n)$, $\ell=2$ for $\mathrm{SU}(n)$, and $\ell=4$ for $\mathrm{Sp}(n)$. In particular, if $\nu\geqslant e^{-cn}$, then the diameter is at most $O_c(1)$. This concerns uniform diameter bounds for large measurable subsets of the classical compact groups; the excerpt gives no resolution.