Borcea's univalence conjecture for polynomial critical points

Let $F$ be an $n$th-degree polynomial with $F(0)=0$, let $F'$ be its derivative, and let $\sigma_p(F)$ denote the $p$-variance of the zeros of $F$. A polynomial is univalent in a disk if it is one-to-one there.

Borcea's univalence conjecture. For $p\geq1$, $F$ is not univalent in any closed disk centered at zero whose radius is larger than

$$\sigma_p(F)\sin\left(\frac{\pi}{n}\right).$$

This is motivated by the Alexander–Kakeya theorem, whose extremal polynomials are of the form $a(z-c)^n-b$. The source presents the proposed strengthening as a conjecture and gives no resolution.