The max-plus eigenspace invariance conjecture for Perron-Frobenius asymptotics

Let $A$ and $B$ be matrices, let $\sigma_{(\max+)}(A)$ and $\sigma_{(\max+)}(B)$ denote their max-plus eigenspaces, and let $\mathscr{P}_{\infty}(A)$ and $\mathscr{P}_{\infty}(B)$ denote the corresponding limits of normalized Perron eigenvectors. Max-plus eigenspace invariance conjecture. If

$$\sigma_{(\max+)}(A)=\sigma_{(\max+)}(B),$$

then

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

This is presented as the second conjecture, as a generalization supported by the paper's experiments. It asserts that the asymptotic Perron-Frobenius limit depends only on the max-plus eigenspace, but no proof or resolution is given.