Eigenvalue model conjecture for volume distortion

Let $M$ be a matrix with at least one eigenvalue off the unit circle, and consider the semidirect product $\mathbb{Z}^k\rtimes_M\mathbb{Z}$. Let $N$ be the diagonal matrix whose diagonal entries are the norms of the eigenvalues of $M$. Eigenvalue model conjecture. The distortion of $\mathbb{Z}^k$ in $\mathbb{Z}^k\rtimes_M\mathbb{Z}$ is the same as that of $\mathbb{R}^k$ in $\mathbb{R}^k\rtimes_N\mathbb{R}$. The conjecture asserts that the exponential changes produced by the eigenvalues dominate the polynomial changes from nontrivial Jordan blocks, so the latter do not affect the large-scale distortion.