(4,6)-cage occupancy-fraction conjecture

Let $\alpha_G(\lambda)$ denote the occupancy fraction of a graph $G$ in the hard-core model, and let $H_{4,6}$ be the $(4,6)$-cage. (4,6)-cage occupancy-fraction conjecture. For every $4$-regular graph $G$ of girth at least $5$ and every $\lambda>0$,

$$\alpha_G(\lambda)\leq\alpha_{H_{4,6}}(\lambda).$$

This is the proposed $4$-regular analogue of the corresponding extremal result and remains open.