Nonnegative irreducible matrices, Perron eigenvector and Dietzenbacher's theorem

Question

Let $A\geq 0$ irreducible, and let $\hat{A}\geq 0$ and irreducible obtained by changing some entries of $A$. Consider $Ax=\rho x$ and $\hat{A}\hat{x}=\hat{\rho}\hat{x}$ the Perron's eigencouples, and denote by $\it{I}$ the set of unchanged rows. Prove that if $\rho=\hat{\rho}$ then either $x$ is multiple of $\hat{x}$ or $$\forall i \in \it{I}~ \min_{j=1\dots n}\frac{\hat{x}_j}{x_j}<\frac{\hat{x}_i}{x_i}<\max_{j=1\dots n}\frac{\hat{x}_j}{x_j}$$ P.S.: there's a theorem stating that whithout the hypothesis $\rho=\hat{\rho}$ it holds $$\forall i \in \it{I}~ \frac{{\rho}}{\hat{\rho}}\min_{j=1\dots n}\frac{\hat{x}_j}{x_j}\leq\frac{\hat{x}_i}{x_i}\leq\frac{{\rho}}{\hat{\rho}}\max_{j=1\dots n}\frac{\hat{x}_j}{x_j}$$ Thank you in advance.