The Koebe radius conjecture for polynomials of degree N

Let $p_N$ be a polynomial of degree $N$ in the normalized univalent class $\mathcal S$, and define its Koebe radius by

$$r_N:=\inf\{|p_N(z)|:z=e^{it},\ 0\leq t\leq 2\pi\}.$$

The Koebe radius is the infimum of the boundary modulus among such normalized univalent polynomials. Koebe radius conjecture. The value of the Koebe radius for polynomials of degree $N$ is

$$\frac14\sec^2\frac{\pi}{N+2}.$$

The paper introduces a new family of typically real polynomials conjectured to be univalent and to attain this value; univalence is proved for degrees at most $6$, while the claimed general formula remains open in the supplied text.