Tutte–Coxeter graph occupancy-fraction conjecture

Let $\alpha_G(\lambda)$ denote the occupancy fraction of a graph $G$ in the hard-core model, and let $H_{3,8}$ be the $(3,8)$-cage, also known as the Levi graph or Tutte–Coxeter graph. Tutte–Coxeter occupancy-fraction conjecture. For every $3$-regular graph $G$ of girth at least $7$ and every $\lambda>0$,

$$\alpha_G(\lambda)\leq\alpha_{H_{3,8}}(\lambda).$$

The conjecture identifies the $(3,8)$-cage as the maximizer in this girth-constrained problem and is open.