Elementary equivalence and diffeomorphism groups of smooth manifolds

Let two smooth manifolds $M$ and $N$ be given, with $M$ closed and connected. If the $C^r$--diffeomorphism group of $M$ is elementarily equivalent to the $C^s$--diffeomorphism group of $N$ for some $r,s\in[1,\infty)\cup\{0,\infty\}$, then $r=s$ and $M$ and $N$ are $C^r$--xdiffeomorphic. This strengthens a previously known result by Takens and Filipkiewicz, which implies that a group isomorphism between diffeomorphism groups of closed manifolds necessarily arises from a diffeomorphism of the underlying manifolds, provided the regularity lies in $\mathbb{N}\cup\{\infty\}$. We prove an analogous result for groups of diffeomorphisms preserving smooth volume forms, in dimension at least two.