Conjectured distance threshold for $(4,s)$-Salem sets

Let $\mathbb{F}_q$ be a finite field, let $d\geq 2$, and let $E\subseteq\mathbb{F}_q^d$ be a $(4,s)$-Salem set. Let $\alpha(d,4,s)$ be the smallest exponent such that every $(4,s)$-Salem set $E$ with $|E|\gg q^{\alpha(d,4,s)}$ satisfies $|\Delta(E)|\gg q$. Distance-threshold conjecture for $(4,s)$-Salem sets. One has

$$\alpha(d,4,s)= \begin{cases} 1,& d=2,\ s\in[\tfrac14,\tfrac12],\\[3pt] \frac{d}{2},& d\ge4\ \text{even},\ s\in[\tfrac14,\tfrac{d+2}{4d}],\\[3pt] \frac{d+2}{8s},& d\ge4\ \text{even},\ s\in[\tfrac{d+2}{4d},\tfrac12],\\[3pt] \frac{d+1}{8s},& d\ge3\ \text{odd},\ s\in[\tfrac14,\tfrac12]. \end{cases}$$

The proposed values are supported by constructions of $(4,s)$-Salem sets with few distances and by known distance results, including the Erdős–Falconer threshold in dimension two and sharp results for sets on spheres or arbitrary sets in other dimensions. The stated thresholds remain conjectural in the general $(4,s)$-Salem setting.