Let $M(n,k)$ be the set of $n\times n$ matrices of nonnegative integers such that every row and every column sums to $k$.  Let $P(n,k)$ be the fraction of such matrices which have no zero entries, equivalently the probability that a random matrix from the uniform distribution on $M(n,k)$ has no zero entries.

One thing to note is that $$P(n,k)=\frac{|M(n,k-n)|}{|M(n,k)|}$$
(think about subtracting 1 from every entry).  Also note that $P(n,k)$ is the fraction of integer points in the $k$-dilated Birkhoff polytope that lie in the interior.

It seems "obvious" that $P(n,k)$ is a non-decreasing function of $k$. For large enough $k$ it is strictly increasing by Ehrhart theory, but I'd like to see a proof for all $k$.  So the problem is: 

**Prove that $P(n,k)$ is a non-decreasing function of $k$ for fixed $n$**.