Bourgain–Chang bilinear estimate conjecture over finite fields

Let $p$ be a prime, let $\mathbb{F}_p$ be the finite field with $p$ elements, and define

$$T(f_1,f_2)(s)=\sum_{n\ne s,\,n\in\mathbb{F}_p}f_1(s-n)f_2(n)K(s-n,n),$$

where

$$K(a,b)=\frac{1}{p}\sum_{y\in\mathbb{F}_p}e_p(ay^2+by),\qquad e_p(x)=e^{-2\pi i x/p}.$$

Bourgain–Chang bilinear estimate conjecture. For every $\varepsilon>0$, there exists $C_\varepsilon$ such that, for all sufficiently large primes $p$,

$$\|T(f_1,f_2)\|_2\leq C_\varepsilon p^{\varepsilon-1/2}\|f_1\|_2\|f_2\|_2.$$

This conjecture concerns the expected near-optimal $L^2\times L^2\to L^2$ bound for a bilinear operator whose kernel is a normalized quadratic Gauss sum. The source attributes it to Bourgain and Chang; the paper improves an existing estimate but does not state that this conjecture is resolved.