Schmeisser's strengthening of Sendov's conjecture

Let $F$ be an $n$th-degree polynomial with $n\geq2$, let $V(F)$ be its multiset of zeros, and let $\sigma_\infty(F)$ be the radius of the smallest closed disk containing $V(F)$. Let $\operatorname{conv}(V(F))$ denote the convex hull of the zeros.

Schmeisser's conjecture. For every $\zeta\in\operatorname{conv}(V(F))$, the closed disk centered at $\zeta$ with radius $\sigma_\infty(F)$ contains a critical point of $F$.

This strengthens Sendov's conjecture by allowing the center to be any point in the convex hull rather than a zero of $F$. The source attributes it to Schmeisser and gives no resolution.