Bourgain–Demeter's quadratic Vinogradov mean value conjecture

Let $J_{s,2}(A)$ denote the number of solutions to the quadratic Vinogradov system for a finite set $A$ of real numbers, with $s$ pairs of variables. Let $s\geq 3$ be a natural number, let $A$ be a finite, non-empty set of real numbers, and let $\epsilon>0$ be real. Bourgain–Demeter's conjecture. One has

$$J_{s,2}(A)\ll_{\epsilon}|A|^{2s-3+\epsilon}.$$

The conjecture asserts that the lower bound supplied by the structured sets $A_N=\{1,2,\ldots,N\}$ is sharp up to a factor of $|A|^{\epsilon}$. Its case $s=3$ was posed by Bourgain and Demeter, and the displayed estimate for general $s$ is equivalent to that case via Young's convolution inequality.