The hardcore-model tree spectral-gap conjecture

Let $T$ be a tree with $n$ vertices, and let $\lambda$ be the fugacity of the hardcore model. The Glauber dynamics is the single-site Markov chain for this Gibbs distribution, and its spectral gap is the smallest nonzero eigenvalue of the chain's Laplacian. The hardcore-model tree spectral-gap conjecture. For any tree $T$ with $n$ vertices, the spectral gap of the Glauber dynamics for the hardcore model on $T$ with fugacity $\lambda<e-1$ is at least $\Omega(n^{-1})$. This is proposed from the reconstruction threshold: the source notes that the reconstruction threshold exceeds $e-1$, while rapid mixing is known for sufficiently small fugacity and polynomial lower bounds occur for sufficiently large fugacity. The conjectured threshold and the full arbitrary-tree statement remain open.