Even-dimensional Erdős–Falconer distance conjecture

Let $d\ge 2$ be an even integer and let $A\subseteq\mathbb F_q^d$. For $x,y\in\mathbb F_q^d$, write

$$\|x-y\|=\sum_{i=1}^d(x_i-y_i)^2,$$

and define the distance set $\Delta(A)=\{\|x-y\|:x,y\in A\}$. Here $X\sim Y$ means that $X\gtrsim Y$ and $Y\gtrsim X$, with constants independent of $q$.

Even-dimensional Erdős–Falconer conjecture. If $|A|\ge Cq^{d/2}$ for a sufficiently large constant $C$ independent of $q$, then

$$|\Delta(A)|\sim q.$$

This is presented as an open problem in every even dimension. Partial improvements are known in dimension $2$, including bounds $4/3$ and, over prime fields, $5/4$, but the conjectured exponent $d/2$ is not known in all even dimensions.