Univalence conjecture for the extremal Chebyshev polynomial

Let $N$ be a positive integer, let $U_m$ denote the Chebyshev polynomial of the second kind, and let $U'_m$ denote its derivative. Define

$$F(z)=\frac{1}{U'_N\left(\cos\frac{\pi}{N+2}\right)}\sum_{j=1}^N U'_{N-j+1}\left(\cos\frac{\pi}{N+2}\right)U_{j-1}\left(\cos\frac{\pi}{N+2}\right)z^j.$$

Univalence conjecture. The polynomial $F$ is univalent in the unit disk $\mathbb D$. Numerical evidence supports the assertion for all $N$, while it is established in the cases $N=3$ and $N=4$.