Finite fields distance conjecture

Let $q$ be odd, let $d$ be even, and let $E\subseteq\mathbb{F}_q^d$. Define the distance set

$$D(E)=\left\{\sum_{i=1}^d(x_i-y_i)^2:(x_1,\dots,x_d),(y_1,\dots,y_d)\in E\right\}.$$

Finite fields distance conjecture. If $\#E\geqslant Cq^{d/2}$ with $C$ sufficiently large, then

$$\#D(E)\gtrsim q.$$

This is the finite-fields analogue of the Erdős distinct-distances problem and the Falconer distance problem. The statement concerns whether a sufficiently large set in even-dimensional finite-field space determines a positive proportion of all possible distances; its resolution status is not specified in the supplied text.