Occupancy-fraction conjecture for independent sets in triangle-free graphs

Let $G=(V,E)$ be a triangle-free graph of average degree $d$, let $Z_G(\lambda)$ denote its independence polynomial, and write $Z_G'(1)$ for its derivative at $1$. Occupancy-fraction conjecture. As $d\to\infty$,

$$\frac{Z_G'(1)}{Z_G(1)}\geq (1+o(1))\frac{\log d}{d}|V|.$$

This conjecture proposes an asymptotically sharp lower bound on the expected size of an independent set under the uniform distribution on independent sets. It would strengthen the paper's main theorem essentially by integration and is related to a conjecture of Davies and Kahan; its status is not resolved in the supplied text.