Conjecture on the sharp threshold for embedding subdivisions in finite-field distance graphs

Let $E \subseteq \mathbb{F}_q^d$ and let $R \subseteq \mathbb{F}_q \setminus \{0\}$. Let $C$ be the constant appearing in Theorem, whose conclusion guarantees the existence of the specified $R$-distance subdivision of a complete graph in $E$ when the relevant size condition holds. Sharp-threshold conjecture. The conclusion of Theorem holds for

$$C = O(q^{(d+1)/2}).$$

The preceding discussion says that the exponent $(d+1)/2$ is the natural threshold for finding a single distance edge, and that the dependence on the number of distances is understood, while the dependence on $|R|$ is believed removable. The conjecture asserts that the theorem's additive size threshold can be reduced to order $q^{(d+1)/2}$, independently of the current larger bound.