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My, i.e. hyperbolic polynomials, approach falls a bit short of proving the conjecture: first of all, the bound from my inequality is k^n G(k)^{n-k} \frac{k!}{k^k}, and it is integer only if k=1,2,n. So it can't be the minimum of permanents of integer matrices. Nobody knows the exact value of the minimum for given (k,n), I have not even seen a conjecture on that. This is why sparse problem is so much more interesting than, say, the Van Der Waerden Conjecture. More seriously, my approach actually needs the degrees of variables x_i in the polynomials (\partial_n....\partial_{i+1}) Prod_{A}. There is a simple upper bound on those degree in terms of the sparsity, but it is not sharp. Actually k^n (G(k)^{n-k} \frac{k!}{k^k} is attainable in this general setting, see my last paper in ECCC. Now, back to Ryzer conjecture: Let A be $n \times n$ minimizer. And a(n) be the number of its boolean rows (the same with columns). It follows from my approach + plus the known upper bound due to Schrijver that \lim_{n \rightarrow \infty} \frac{\min_{over minimizers}(a(n))}{n} = 1. BTW, the same applies to mixed discriminants.