Eigenvalue-crossing conjecture for non-planar transfer matrices

Consider a family $G_{n,L}$ of non-planar connected graphs as above. Let $Q_\ell$ be the smallest $Q>0$ for which the transfer matrix $\widetilde{\sf T}_{L,\ell,(\ell)}$ for the fully symmetric representation $(\ell)$ of $\ell$ marked blocks has at least two dominant eigenvalues, or set $Q_\ell=\infty$ if none exists. For $Q\in(0,Q_\ell)$, let $\lambda_\ell$ be the unique dominant eigenvalue. Eigenvalue-crossing conjecture. One has $Q_0<Q_1<Q_2<\cdots$; each $\lambda_\ell$ is real; the equation $|\lambda_\ell|=|\lambda_{\ell'}|$ has at most one solution for $0<Q<\min(Q_\ell,Q_{\ell'})$; and, if a solution exists, it is $Q=\ell+\ell'-1$. The source reports validation for generalized Petersen graphs with $L=k+1$ and $k\leq7$.