Extremizer conjecture for the T-symmetric univalent polynomial problem

Let $T$ and $N$ be positive integers, and consider the optimization problem over real-coefficient univalent polynomials in $\mathbb D=\{z:|z|<1\}$ with $T$-fold symmetry and degree $(N-1)T+1$:

$$F(z)=\sum_{j=1}^{N}a_jz^{T(j-1)+1},\qquad a_1=1.$$

Let $U_m$ denote the Chebyshev polynomial of the second kind. Extremizer conjecture. The extremizer is

$$F^{(0)}(z)=z+\sum_{j=2}^{N}a_j^{(0)}z^{T(j-1)+1},$$

where

$$a_j^{(0)}=\frac{U'_{T(N-j+1)}\left(\cos\left(\frac{\pi}{TN+2}\right)\right)}{U'_{TN}\left(\cos\left(\frac{\pi}{TN+2}\right)\right)}\prod_{k=1}^{j-1}\frac{\sin\left(\frac{\pi(2+T(k-1))}{TN+2}\right)}{\sin\left(\frac{\pi Tk}{TN+2}\right)},\qquad j=2,\ldots,N,$$

and this polynomial is univalent. The conjecture gives an explicit candidate for the extremizer of the generalized $T$-symmetric optimization problem. The paper notes that the associated asymptotics remain to be determined and that numerical images support univalence, but it provides no proof of the conjecture.