The monomer-dimer spectral independence conjecture

Let $G=(V,E)$ be a 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 monomer-dimer spectral independence conjecture. For any 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 conjecture is motivated by the bounded spectral independence observed on regular trees despite their unbounded total influence; it is false for multigraphs with parallel edges, so the simple-graph refinement below is the unresolved form.