Optimality of the constructed polynomials for $J_N^{(T)}$

Let $N$ and $T$ be positive integers, and let $J_N^{(T)}$ denote the optimization problem considered for Case B. Let $q(z)$ be the constructed polynomial with parameters $\sigma=\tau=1$.

Optimality conjecture. For any $N$ and $T$ the polynomials $q(z)$ with $\sigma=\tau=1$ give the solutions to the optimization problem $J_N^{(T)}$.

If true, this would identify the proposed filter polynomials as solutions of the Case B optimization problem and support their use for detecting high-order cycles. The source gives no resolution of the conjecture.