I have posted a related question before (link), but I didn't really get a completely satisfactory answer, and also I believe that I was able to simplify the problem a little. Therefore, I hope that it is appropriate to post again.
Basically, I want to show that
$f(y,z) := 81 y^4+\left(450 y^2+228 y^3+476 y^4\right) z+\left(405+108 y+1692 y^2+504 y^3+884 y^4\right) z^2+\left(1404-168 y+1828 y^2-144 y^3+528 y^4\right) z^3+\left(1872-624 y+672 y^2-576 y^3\right) z^4+\left(1232-192 y+320 y^2\right) z^5+320 z^6$
is positive for any $y > 0, z>0$.
By inspection of the $y$ polynomials, it is easy to see that only the polynomial before $z^4$, $g(y) := 1872-624y + 672y^2-576y^3$, might cause troubles for $y$ sufficiently large. However, even when ignoring the positive terms in $g(y)$, and taking $h(y) := -624y - 576y^3$ instead, numerically it still holds that $f(y,z)$ is strictly positive.
It hope that there is some easy way to show positivity of $f(y,z)$. Any idea would be greatly appreciated. Thanks!