Richards' static Gauss–Lucas conjecture

For a bounded convex set $K\subset\mathbb{C}$, an $\epsilon>0$, and a polynomial $p$, let $K_\epsilon$ be the $\epsilon$-neighborhood of $K$ and let $Z(p,K)$ denote the number of zeros of $p$ in $K$. Richards' static Gauss–Lucas conjecture. For every bounded convex set $K\subset\mathbb{C}$ and every $\epsilon>0$, there is a constant $C_{K,\epsilon}\in(0,1)$ such that, for every polynomial $p$ of sufficiently large degree, if

$$\frac{Z(p,K)}{\deg(p)}>C_{K,\epsilon},$$

then

$$Z(p',K_\epsilon)\geq Z(p,K)-1.$$

This conjecture proposes a finite-degree principle underlying the asymptotic Gauss–Lucas theorem: if sufficiently many zeros of a high-degree polynomial lie in a convex set, then nearly as many critical points lie in every prescribed neighborhood of that set. Its resolution is not given in the source.