The refined graph-container bound at activity of order inverse degree

Let $d_X$ and $d_Y$ be sufficiently large integers, let $\delta\geq 1$ and $\delta',\delta”>0$, and let $\Sigma=X\sqcup Y$ be a $\delta$-approximately $(d_X,d_Y)$-biregular graph satisfying the assumptions of Lemma ML. Write $\mathcal{G}(a,g)$ for the family of sets occurring there and let $\lambda$ be the hard-core activity. The refined graph-container conjecture. A bound of the type in Lemma ML should hold for $\lambda=\widetilde{\Omega}(1/d_X)$; equivalently, there should be a constant $\kappa$ and a function $\lambda^\ast:\mathbb{N}\to\mathbb{R}$ with $\lambda^\ast(d)=\widetilde{\Omega}(1/d)$ as $d\to\infty$ such that, under the assumptions of Lemma ML, if $\lambda\geq\lambda^\ast(d_X)$, then

$$\sum_{A\in\mathcal{G}(a,g)}\lambda^{|A|}\leq |Y|(1+\lambda)^g\exp\left\{-(g-a)/d_X^{\kappa}\right\}.$$

The proved lemma only establishes the corresponding estimate for $\lambda$ above a constant multiple of $\log^2 d_X/d_X^{1/2}$, and the conjecture would extend the useful range to essentially inverse degree; the authors state that their current methods encounter a barrier at this point.