Davies–Jenssen–Perkins–Roberts conjecture on independent sets in triangle-free graphs

For a graph $G$, let $\alpha(G)$ be its independence number and let $\alpha_G(1)$ denote the occupancy fraction of the hard-core model at fugacity $1$. Let $G$ be triangle-free with minimum degree $d$.

Davies–Jenssen–Perkins–Roberts conjecture. As $d\to\infty$,

$$\frac{\alpha(G)}{\alpha_G(1)|V(G)|}\ge 2-o_d(1).$$

If true, this would imply the improved asymptotic bound $R(3,k)\lesssim k^2/(2\log k)$, beyond the constant supplied by the standard local-occupancy approach.