Can a upper bound of $\sum_{b=1}^{p-1}\left(\frac{b^2-a^2}{p}\right)\left(\frac{b^2-1}{p}\right)$, $a\in{Z}$, be strictly less than $\sqrt{p}$?

Question

By weil estimate I can only say, one bound is $\sqrt{p}$. Can it be strictly less than $\sqrt{p}$? I want to see whether one better bound be given or not.