The fugacity-uniform triangle-free average-to-maximum ratio conjecture

Let $G$ be a triangle-free graph. For $4\lambda>04$, let $4\overline{\alpha}_G(\lambda)4$ denote the average size of an independent set in the hard-core model at fugacity $4\lambda4$, and let $4\alpha(G)4$ be the maximum independent set size.

Fugacity-uniform ratio conjecture. For every $4\epsilon>04$, there exists $4\lambda>04$ such that, for every triangle-free graph $G$,

$$\frac{\alpha(G)}{\overline{\alpha}_G(\lambda)} \ge 2-\epsilon.$$

This would yield the same asymptotic improvement to the upper bound on $R(3,k)$ as the minimum-degree conjecture. Its status is not resolved in the supplied text.