The smooth-volume-preservation conjecture for generic random groups

Let $\Gamma$ be a group that is generic in an appropriate model for random groups, let $M$ be a compact manifold, and consider an action of $\Gamma$ on $M$. A smooth volume form is a nowhere-vanishing smooth top-degree differential form on $M$. Smooth-volume-preservation conjecture. Any action of a generic group $\Gamma$ on a compact manifold $M$ preserves a smooth volume form. This is proposed as a concrete statement that would reduce Gromov's conjecture to the paper's main theorem. The appropriate model for random groups is not specified, and the conjecture remains open in the source.