Consider $2n$ vertex balanced bipartite graph.

If total number of edges in it is $n^2$ then we have $n!$ perfect matchings.

Fix $c\in(0,\frac12)$ and consider collection of $2n$ vertex balanced bipartite graphs with at least $cn!$ perfect matchings. If we pick a random graph from this collection then should we expect at least $f(c)n^2$ total number of edges in it for some function $f:\Bbb R_{>0}\rightarrow\Bbb R_{>0}$?