Log-convexity conjecture for bipartite graphical degree-sequence counts

For each positive integer $n$, let $a_n$ denote the number of bipartite graphical degree sequences on two parts of size $n$, written as $n+n$ vertices. Log-convexity conjecture. The sequence $a_n$ is log-convex as a function of $n$, that is, for all relevant $n$, $a_n^2\leq a_{n-1}a_{n+1}$. If proved, this would yield a slightly stronger counting result for bipartite degree sequences admitting rapidly mixing Markov-chain processes; the source gives no resolution of the conjecture.