Gowers's intersection conjecture for approximate quadratic varieties

Let $p\geq 3$ be prime, let $G$ be a finite-dimensional vector space over $\mathbb{F}_p$, and let $V\subseteq G$ be a $(c_0,\delta,\varepsilon)$-approximate quadratic variety, meaning that $|V|=\delta|G|$, $\|\mathbbm{1}_V-\delta\|_{\mathsf{U}^2}\leq\varepsilon$, and

$$\mathop{\mathbb{E}}_{x,a,b,c} \Delta_{a,b,c}\mathbbm{1}_V(x)=c_0\delta^7.$$

Gowers's conjecture. If $\varepsilon$ is sufficiently small in terms of $\delta$ and $c_0$, then there exists a quadratic variety $Q$ such that

$$|Q|\leq O_{c_0}(|V|)\quad\text{and}\quad |Q\cap V|\geq\Omega_{c_0}(|V|).$$

The conjecture predicts that an approximate quadratic variety has substantial overlap with an exact quadratic variety of comparable size. It is presented as an open conjecture, with the resemblance to the Balog--Szemerédi--Gowers/Freiman phenomenon offered as motivation.