Convergence of powers of a dual number matrix

Let $A=A_s+A_d\varepsilon$ be an $n\times n$ dual number matrix and let $k\in\mathbb{N}$. Here $O$ denotes the zero matrix, and $\rho(A_s)$ denotes the spectral radius of the standard-part matrix $A_s$. Power-convergence conjecture.

$$\lim_{k\to\infty}A^k=O$$

if and only if $\rho(A_s)<1$.

The paper states that this conjecture receives a positive answer in the subsequent section, so the claim is presented as resolved by the paper. It concerns the asymptotic behavior of powers of dual number matrices and supplies the convergence result used in the later analysis of the Collatz method.