The simple-graph monomer-dimer spectral independence conjecture

Let $G=(V,E)$ be a simple graph, let $\lambda>0$, and let $\mu$ be the Gibbs distribution of the monomer-dimer model on $G$ with fugacity $\lambda$. The quantity $\lambda_{\max}(\Psi_\mu)$ denotes the largest eigenvalue of its influence matrix. The simple-graph monomer-dimer spectral independence conjecture. For any simple graph $G=(V,E)$ and any $\lambda>0$, the Gibbs distribution $\mu$ of the monomer-dimer model on $G$ with fugacity $\lambda$ has $\lambda_{\max}(\Psi_\mu)=O_{\lambda}(1)$. The large-girth theorem proves the claim for sufficiently large girth, while the parallel-edge example rules out the unrestricted multigraph version. The cases of simple graphs with intermediate girth remain unresolved.