Conjecture on the tightness of the two lower bounds for integral point sets over $\mathbb{Z}_n^2$

Let $\mathcal{I}(n,2)$ denote the maximum number of points in an integral point set over $\mathbb{Z}_n^2$. The paper gives two lower bounds for this quantity, one in Lemma 1 and one in Lemma 2. Tightness conjecture. For all $n\in\mathbb{N}$, either the lower bound of Lemma 1 or the lower bound of Lemma 2 is tight. This conjecture asserts that one of the two stated constructions always attains the maximum; the source provides no evidence of a resolution.