Necklace character-sum conjecture for localized Paley graphs

Let $p$ be prime, let $\chi:\mathbb F_p\to\{-1,0,1\}$ be the Legendre symbol, and for nonempty subsets $Z_1,\ldots,Z_k\subseteq\mathbb F_p$ define

$$\Sigma(Z_1,\ldots,Z_k)=\sum_{x_1,\ldots,x_k\in\mathbb F_p}\chi(x_2-x_1)\cdots\chi(x_k-x_{k-1})\chi(x_1-x_k)\prod_{i=1}^k\prod_{z\in Z_i}\chi(x_i-z).$$

Necklace character-sum conjecture. For every $k\geq1$,

$$\lim_{p\to\infty}p^{-(k/2+1)}\max_{\substack{Z_1,\ldots,Z_k\subseteq\mathbb F_p\Z_i\neq\varnothing\\|Z_1\cup\cdots\cup Z_k|\leq a}}|\Sigma(Z_1,\ldots,Z_k)|=0.$$

These estimates are designed to imply the weak-convergence conjecture for localized Paley graph spectra. They are stated as a stronger analytic input and remain open in the general degree-$a$ setting.