Static approximate Gauss–Lucas conjecture for rational functions

Let $K\subset\mathbb{C}$ be bounded and convex, and let $\epsilon>0$ be given. There is some constant $C>0$ such that if $k/(n-k)>C$, then $K$ satisfies $\mathcal{RAGL}(n,k,\epsilon)$. Static approximate Gauss–Lucas conjecture. A sufficiently large ratio of zeros in $K$ to zeros and poles outside $K$ should guarantee the rational approximate Gauss–Lucas property for $K$, with the threshold depending only on $K$ and $\epsilon$. This is proposed as a static principle underlying Totik's asymptotic approximate Gauss–Lucas theorem; its resolution is not given here.