Parameterized occupancy-fraction conjecture for 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 let $W$ be the Lambert $W$-function, defined by $W(x)e^{W(x)}=x$. Parameterized occupancy-fraction conjecture. For every $\lambda\in(0,1]$,

$$\frac{\lambda Z_G'(\lambda)}{Z_G(\lambda)}\geq \frac{W(\lambda d)-W(2\lambda)}{d-2}|V|.$$

The paper presents this as a more general and precise restatement of the preceding occupancy-fraction conjecture, motivated by differentiating its lower-bound inequality. The supplied text gives no resolution, so the conjecture is recorded as open.