Uniform exactness conjecture for torsion-free hyperbolic and linear groups

Let $\Gamma$ be a group, let $\mathcal{U}$ be a free ultrafilter on $\mathbb{N}$, and write $\Gamma^\mathcal{U}$ for its ultrapower. A group is uniformly exact when the action of $\Gamma^\mathcal{U}$ on the ultrapower $(\ell_\infty\Gamma)^\mathcal{U}$ is amenable. Uniform exactness conjecture. Torsion-free hyperbolic groups and linear groups are uniformly exact. This would provide broad classes of groups whose ultrapower actions have the amenability property defining uniform exactness; the surrounding discussion notes that exactness of the ultrapower is known for these groups, but does not establish the stronger uniform exactness assertion.