Odd conjugate-polynomial extremal-value conjecture

Let

$$C(t)=\sum_{j=1}^N a_j\cos((2j-1)t),\qquad S(t)=\sum_{j=1}^N a_j\sin((2j-1)t),$$

where the coefficients $a_j$ are real and $a_1=1$. Odd conjugate-polynomial conjecture. One has

$$\inf_{a_j}\max_t\{\lvert C(t)\rvert:S(t)=0\}=-\frac12\sec^2\frac{\pi}{2N+2},$$

and the solution is unique, namely

$$a_j^{(0)}=\frac{U'_{2(N-j+1)}\left(\cos\frac{\pi}{2N+2}\right)}{U'_{2N}\left(\cos\frac{\pi}{2N+2}\right)},\qquad j=1,\ldots,N.$$

This is an extremal problem for conjugate trigonometric polynomials; the source gives the corresponding Suffridge construction as a weaker comparison, but does not resolve the conjecture.