Rational-function extension of the static approximate Gauss–Lucas conjecture

Let $K\subset\mathbb{C}$ be convex, and let $\epsilon>0$ be given. There exists some constant $C>0$ for which the following holds. Let $g,h\in\mathbb{C}(z)$ be rational functions, and assume that all zeros, poles, and critical points of $g$ lie in $K$. Rational-function extension conjecture. If

$$\frac{\#_z(g,\mathbb{C})+\#_p(g,\mathbb{C})}{\#_z(h,\mathbb{C})+\#_p(h,\mathbb{C})}>C,$$

then

$$\#_c(gh,K_\epsilon)\geq\#_c(g,K).$$

This extends the preceding static principle to an unbounded convex set and rational functions, assuming that the zeros, poles, and critical points of $g$ already lie in $K$. The supplied text presents it as an additional conjecture and gives no resolution.