The Perron eigenvector conjecture for the O(1) loop-model Hamiltonian

Let $H$ be the Hamiltonian of the O(1) loop model acting on the link-pattern basis, and let $\Psi$ be the vector whose components are the numbers $A_n(\pi)$ of fully packed loop states with link-pattern $\pi$. The largest eigenvalue of $H$ is $2n$.

Perron eigenvector conjecture. The vector $\Psi$ is the unique eigenvector of $H$ corresponding to its largest eigenvalue $2n$.

The source explains that the column sums of $H$ equal $2n$, so its spectral radius is $2n$, and that nondegeneracy follows from positivity of the corresponding transfer matrix. The conjectural content is the identification of the combinatorial vector $\Psi$ with this unique maximal-eigenvalue eigenvector; the source does not provide a general proof.