Koh–Shen even-dimensional finite-field distance conjecture

Let $\mathbb F_q$ be a finite field of characteristic greater than two, let $d\geq 2$ be even, and let $E,F\subset\mathbb F_q^d$. Define

$$\mathcal D(E,F)=\{\lVert x-y\rVert:x\in E,y\in F\},\qquad \lVert z\rVert=z_1^2+\cdots+z_d^2.$$

Koh–Shen conjecture. If $|E||F|\gg_C q^d$, then

$$|\mathcal D(E,F)|\gg_c q.$$

This generalizes the finite-field Falconer distance conjecture to pairs of sets in even dimensions; the supplied text gives no resolution.