Generic transitivity for pairs of volume-preserving diffeomorphisms

Let $M$ be a compact manifold and let $r\in\mathbb N\cup\{\infty\}$. An iterated function system consists of a finite family of homeomorphisms of $M$; it is transitive if the semigroup it generates has a dense forward orbit. Generic transitivity conjecture. For any $r\in\mathbb N\cup\{\infty\}$, a $C^r$-generic pair of volume-preserving diffeomorphisms generates a transitive iterated function system. The claim concerns generic transitivity in the volume-preserving setting; the supplied text gives no resolution status.