A countable discrete group $\Gamma$ is said to be exact if it admits an amenable action on some compact space.

So clearly amenable groups are exact, but large familes of non-amenable groups are as well.

For many of the families that I know of (ex. linear groups, hyperbolic groups) that are exact, they also satisfy the von Neumann conjecture (i.e. that if they are non-amenable then they have subgroup isomorphic to a free group.)

So my questions is:

Are there examples of exact groups that are non-amenable and do not contain free subgroups?