Schoenberg's inequality for polynomial zeros and critical points

Let $F$ be a complex polynomial of degree $n\geq 2$ with zeros $z_1,\dots,z_n$ and critical points $w_1,\dots,w_{n-1}$. Schoenberg's conjecture.

$$\sum_{j=1}^{n-1}|w_j|^2\leq \frac{1}{n^2}\left|\sum_{k=1}^n z_k\right|^2+\frac{n-2}{n}\sum_{k=1}^n|z_k|^2,$$

with equality if and only if all $z_1,\dots,z_n$ are collinear in the complex plane. The conjecture was confirmed independently by Pereira and Malamud, and later reproved by Cheung--Ng and Kushel--Tyaglov.