I'd appreciate any suggestions for how to prove (or disprove) the inequality described below. Some notation first: for positive integers $k$ and $M$, let ${\mathcal D}_{k,M}$ denote the set of all $k \times k$ non-negative integer matrices with row and column sums all equal to $M$. For $k \times k$ matrices $A = [a_{i,j}]$ and $B = [b_{i,j}]$, we write $A \le B$ if $a_{i,j} \le b_{i,j}$ for all $i,j$.
Now, let $M, N$ be fixed positive integers with $M < N$. I'd like to prove that for any $B \in \mathcal{D}_{k,N}$, we have $\displaystyle \sum_{A \in \mathcal{D}_{k,M}: A \le B} \prod_{i,j} \frac{\binom{b_{i,j}}{a_{i,j}}}{\binom{N-b_{i,j}}{M-a_{i,j}}} \ge {\binom{N}{M}}^{2k-k^2}$.
It is easy to check that the inequality holds with equality when $B$ is $N$ times a permutation matrix. It is also straightforward to show that the inequality holds when $B$ has at most two non-zero entries in each row. In particular, the inequality holds when $B$ is a convex combination of $B_1$ and $B_2$, where each of $B_1$ and $B_2$ is $N$ times a permutation matrix. How to proceed when $B$ is a convex combination of three or more such matrices is not clear.