Higher-dimensional Gersten conjecture for volume undistortion

Let $M$ be a square matrix of size at least $m$ with nonzero determinant, let $k<m$, and write $\Gamma_M=\mathbb{Z}^m\rtimes_M\mathbb{Z}$. The subgroup $\mathbb{Z}^m$ is $k$-volume undistorted when its $k$-volume distortion function is linear. Higher-dimensional Gersten conjecture. The group $\mathbb{Z}^m$ is $k$-volume undistorted in $\Gamma_M$ if and only if $M$ has finite order. This proposes a higher-dimensional analogue of Gersten's area-undistortion conjecture; the preceding examples show that volume distortion can depend substantially on the dimension.