An entropy inequality for the bi-multivariate hypergeometric distribution

Given parameters $\bar r = {r_1}, \ldots ,{r_m}$ and $\bar c = {c_1}, \ldots ,{c_n}$ with $\sum {{r_i}} = \sum {{c_j}} = N$, the bi-multivariate hypergeometric distribution is the distribution on nonnegative integer $m \times n$ matrices with row sums $\bar r$ and column sums $\bar c$ defined by ${\text {Prob}}\left ( A \right ) = \prod {{r_i}} !\prod {{c_j}} ! / \left ( {N!\prod {{a_{ij}}!} } \right )$. It is shown that the entropy of this distribution is a Schur-concave function of the block-size parameters.