The entropy lower-bound conjecture for graphs with a given degree sequence

Let $D$ be a strict graphic sequence of type $\varepsilon$, and let $H_{1}(x)$ be the entropy function whose maximum is attained at $\widetilde{\mathbf{p}}$. Write $\mathbb{G}^{D}$ for the set of graphs with degree sequence $D$. Entropy lower-bound conjecture. There exists a number $\eta>0$ independent of $n$ such that

$$e^{-\eta n\cdot\log(n)}\cdot e^{H_{1}(\widetilde{\mathbf{p}})}\leq\left|\mathbb{G}^{D}\right|\leq e^{H_{1}(\widetilde{\mathbf{p}})}.$$

The upper bound is the easy part and follows from the entropy estimate, while the lower bound is explicitly identified as open. This conjecture would provide the needed estimate for the number of graphs with the prescribed degree sequence.