$\newcommand\Legendre{\genfrac(){}{}}$Let prime number $p\equiv 1\pmod 4$,and $x_{i}\ge 0$ such $$x_{1}+x_{2}+\dotsb+x_{p}=1.$$ Show that $$\sum_{1\le i<j\le p}\Legendre{i-j}{p}x_{i}x_{j}\le\dfrac{p-1}{2p+6}$$

Here $\Legendre\cdot\cdot$ is the Legendre symbol.

This problem is an inequality that my colleague encountered when he was writing this paper,and We are can't prove this inequality , so I ask it,and we found the right constant seems to be the best and we found that $p = 5$ can be reached: $$p=5,x_{1}=x_{2}=0.5,x_{3}=x_{4}=\dotsb=0.$$