Second-moment conjecture for matchings in random biregular graphs

Let $G$ be the random $(a,b)$-biregular bipartite graph considered in the paper, and let $m_k(G)$ denote the number of matchings of size $k$. Second-moment conjecture. There exists a constant $C$, independently of $n$ and $k$, such that

$$\mathbb{E}m_k(G)^2\leq C\bigl(\mathbb{E}m_k(G)\bigr)^2.$$

This conjectured second-moment bound would imply the lower bound $\lambda_{\mathbb{T}_{a,b}}(p)\geq \mathbb{G}_{a,b}(p)$ for the matching entropy of the infinite $(a,b)$-biregular tree. The source presents it as a proposed conjecture and gives no resolution.