Davies et al.'s variance lower-bound conjecture for the hard-core model

Let $G$ be an $n$-vertex graph, and let $V_G(\lambda)$ denote the variance parameter of the hard-core model on $G$ at fugacity $\lambda$. Write $K_m$ for the complete graph on $m$ vertices, and let $\Delta$ be the maximum degree of $G$.

Davies et al.'s variance conjecture. For any $\lambda>0$,

$$V_G(\lambda)\geq V_{K_n}(\lambda).$$

If $G$ has maximum degree $\Delta$, then the stronger bound

$$V_G(\lambda)\geq V_{K_{\Delta+1}}(\lambda)$$

should hold.

This conjecture proposes the complete graph as the minimizer of the variance parameter, with a sharper comparison determined by the maximum degree. The source reports it as a conjecture posed by Davies et al.; its resolution is not specified here.