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 $n$.

COMMENT: By a result of Stanley (see David Speyer's answer for refs) there are non-negative integers $\{h_i\}$ such that $$|M(n,k)| = \sum_{i=0}^d h_i \binom{k+i}{d}$$ where $d=(n-1)^2$. I'm wondering if this is enough. Does every polynomial of that form have the desired properties? If so, the result will generalise a lot since every lattice polytope has such an enumerator. Careful: some of the binomial coefficients are 0 when $k\lt d$.