Gersten's area-undistortion conjecture for abelian-by-cyclic groups

Let $M\in GL(k,\mathbb{Z})$ and let $k\geq 3$. Consider the semidirect product $\mathbb{Z}^k\rtimes_M\mathbb{Z}$, where $M$ acts on $\mathbb{Z}^k$. The subgroup $\mathbb{Z}^k$ is area undistorted when its area distortion is linear. Gersten's conjecture. The group $\mathbb{Z}^k$ is area undistorted in $\mathbb{Z}^k\rtimes_M\mathbb{Z}$ if and only if $M$ has finite order in $GL(k,\mathbb{Z})$. This generalizes the known fact that the copy of $\mathbb{Z}^2$ is always area-undistorted in $\mathbb{Z}^2\rtimes_M\mathbb{Z}$.