The bilinear distance conjecture

Let $Q_0,Q_1,Q_2$ be three cubes in $\mathbb{B}(0,C)$ of radius $\approx1$ satisfying

$$|(x_1-x_0)\wedge(x_2-x_0)|\approx1$$

for all $x_0\in Q_0$, $x_1\in Q_1$, and $x_2\in Q_2$. For each $j=0,1,2$, let $E_j$ be a $(\delta,1)_2$ subset of $Q_j$, and let $D$ be a $(\delta,1/2)_1$ subset of $\mathbb{R}$. Bilinear distance conjecture. There is an absolute constant $c_1>0$ such that

$$\left|\{(x_0,x_1,x_2)\in E_0\times E_1\times E_2:|x_0-x_1|,|x_0-x_2|\in D\}\right|\lessapprox\delta^{3-c_1}.$$

The source presents this bilinear estimate as a way to overcome the counterexample obstructing the naive discretized distance problem; a positive answer is stated to imply the distance-set improvement conjecture.