Exact value of the normalized integral-point-set number when $-1$ is a square

Let $\mathbb{F}_q$ be a finite field, let $\square_q$ denote the set of squares in $\mathbb{F}_q$, and let $\overline{\mathcal{I}}(\mathbb{F}_q,2)$ denote the normalized maximum size of an integral point set in the affine plane over $\mathbb{F}_q$. Exact-value conjecture. If $-1\in\square_q$, then

$$\overline{\mathcal{I}}(\mathbb{F}_q,2)=\frac{q-1}{2}.$$

The preceding corollary establishes only the bounds $\frac{q-1}{2}\le\overline{\mathcal{I}}(\mathbb{F}_q,2)\le\frac{q+3}{2}$ in this case, so the asserted equality sharpens the upper bound and is not resolved in the supplied text.