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][1]. 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.


  [1]: http://en.wikipedia.org/wiki/Permanent#Van_der_Waerden.27s_conjecture