Schmeisser's conjecture: every point of the zeros' convex hull is within distance $1$ of a critical point

Let $p$ be a complex polynomial of degree at least $2$ all of whose zeros lie in the closed unit disk. Schmeisser's conjecture asserts that every point of the convex hull of the zeros lies within distance $1$ of some critical point of $p$:

This strengthens the Sendov conjecture, which is the special case $\zeta = z_j$ for a zero $z_j$: every zero of such a polynomial has a critical point within distance $1$. Since each zero belongs to the convex hull of the zeros, Schmeisser's statement implies Sendov's, and it is strictly stronger because the hull generally contains points far from every individual zero. Sendov's conjecture is already on this platform as sendov_conjecture; a proof of the present statement would settle it.

The bound $1$ is sharp and is attained: for $p(z) = z^n - 1$ the critical points all coincide at the origin, which is the centroid of the zeros and lies at distance exactly $1$ from each of them. Numerical search over random polynomials of degree $2$ through $8$ finds no violation, with the worst observed hull distance below $0.99$, and the family $z^n - 1$ attaining $1$ exactly at $n = 2, 3$.

Formalization note. The hypothesis constrains only the zeros of $p$; the degree bound $2 \le \deg p$ rules out the constant and linear cases, where the derivative has no root and the conclusion would fail vacuously in the wrong direction. The convex hull is taken over the reals, which is the correct notion for $\mathbb{C}$ viewed as a real plane.