Conjecture on unbounded minimum cardinalities of maximal integral point sets

Let $q$ be a prime power, and let $\mathcal{A}_{q,s}$ denote the number of non-isomorphic inclusion-maximal integral point sets over $\mathbb{F}_q^2$ having cardinality $s$. Write $l_q$ for the minimum cardinality occurring among such sets.

Unbounded-minimum conjecture. For each $w\in\mathbb{N}$ there exists $q_w\in\mathbb{N}$ such that

$$l_q\ge w$$

for $q\ge q_w$; equivalently,

$$\mathcal{A}_{q,s}=0$$

for $s<w$ and $q\ge q_w$.

This predicts that the minimum size of an inclusion-maximal integral point set tends to infinity with the field size, extending the paper's finite computational observations.