The critical-eigenvector conjecture for tropical Perron-Frobenius asymptotics

Let $A$ be a matrix with max-plus eigenspace $\sigma_{\max+}(A)$, and let $\mathscr{P}_{\infty}(A)$ denote the limit of the normalized Perron eigenvectors of the associated matrices $A_k$. Let

$$v=(0,v_1,v_2)\in\sigma_{\max+}(A).$$

Critical-eigenvector conjecture. If, for every other $w=(0,w_1,w_2)\in\sigma_{\max+}(A)$, there exists an $\alpha>0$ with $\alpha\in\mathbb{R}$ such that $v_1=w_1+\alpha$ and $v_2=w_2+\alpha$, then

$$v=\mathscr{P}_{\infty}(A).$$

The conjecture identifies the asymptotic normalized Perron eigenvector when one critical eigenvector is uniformly translated relative to every other point of the max-plus eigenspace. It is stated without proof and is supported by the experimentation described in the paper; the author notes that it appears to generalize to $n\times n$ matrices.