Linear convergence of the Collatz method for dual number matrices

Let $A=A_s+A_d\varepsilon$ be a primitive dual number matrix, and let $\lambda$ be its Perron eigenvalue. Starting from a dual number vector with positive standard part, define the Collatz iterates by

$$\mathbf{x}^{k+1}=\frac{\mathbf{y}^{k}}{\|\mathbf{y}^{k}\|_2},\qquad \mathbf{y}^{k+1}=A\mathbf{x}^{k+1},$$

and define

$$\underline{\lambda}_k=\min_i\frac{(A\mathbf{x}^k)_i}{x_i^k},\qquad \overline{\lambda}_k=\max_i\frac{(A\mathbf{x}^k)_i}{x_i^k}.$$

Collatz convergence-rate conjecture. What is the condition such that both sequences $\{\underline{\lambda}_k\}$ and $\{\overline{\lambda}_k\}$ converge to the same dual number $\lambda$, and what is the convergence rate?

The monotonicity bounds are known, and the standard parts of the two sequences are known to converge linearly to the same number. The conjecture concerns convergence of the full dual-number sequences and the conditions and rate required for that convergence; the paper states that it gives a positive answer in the following analysis.