I am wondering if the following generalization of van der Waerden's conjecture is true.

Suppose A is an n x n non-negative matrix with all column sums equal to 1, and the sum of row i equal to $T_i$. Then $per(A) \geq T_1\ldots T_n \frac{n!}{n^n}$. This obviously implies van der Waerden's conjecture. I can check it by hand for 2x2 matrices, and I did not have the patience to try larger examples (so it may be false for some easy example). I couldn't modify Gurvits's proof to work either.