Friedland's Lower Matching Conjecture

Let $G$ be a $d$-regular bipartite graph on $v(G)=2n$ vertices, and let $m_k(G)$ denote the number of matchings of size $k$. Put $p=\frac{k}{n}$. Friedland's Lower Matching Conjecture.

$$m_k(G)\geq {n \choose k}^2\left(\frac{d-p}{d}\right)^{n(d-p)}(dp)^{np}.$$

The case $p=1$ is Schrijver's theorem on perfect matchings, and Gurvits proved an asymptotic version. The paper proves this conjecture, in fact obtaining a slightly stronger bound with an extra $c_p\sqrt{n}$ factor when $p$ is separated from $0$ and $1$; it is therefore solved.