Uniqueness conjecture for powers of one minus cosine

Let a trigonometric polynomial of order $n$ mean an object in the class used by the paper, and let condition (5.59) be the condition referred to in the source. Define

$$P_n(\theta)=(1-\cos\theta)^n.$$

Uniqueness conjecture. $P_n(\theta)$ is the only trigonometric polynomial of order $n$ satisfying (5.59). The claim proposes uniqueness within that class, beyond the explicitly verified low-order cases mentioned in the surrounding discussion.