Uniqueness conjecture for companion zeros of the Fejér-Dirichlet indicator

Let $q>1$ be fixed and let $p\geq 5$ be an odd prime. Theorem~ provides a real zero of $\mathfrak F(\cdot,q)$ in $(p-1,p)$. Uniqueness conjecture. For every such $q$ and $p$, this zero is unique. This conjecture concerns the real-zero geometry of the original indicator $\mathfrak F$ between consecutive integers; the supplied text establishes existence but gives no evidence that uniqueness has been proved or disproved.