Conjecture on the cardinality spectrum for q congruent to 3 modulo 4

Let $q$ be a prime power with $q\equiv 3\pmod 4$. Let $\mathcal{A}_{q,s}$ denote the number of non-isomorphic inclusion-maximal integral point sets over $\mathbb{F}_q^2$ having cardinality $s$.

Cardinality-spectrum conjecture. There exist $l_q,r_q\in\mathbb{N}$ such that

$$r_q\le\frac{q-1}{2},$$

$\mathcal{A}_{q,l_q}>0$, $\mathcal{A}_{q,r_q}>0$, $\mathcal{A}_{q,\frac{q+3}{2}}>0$, $\mathcal{A}_{q,q}>0$, and

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

for $s\notin\left\{l_q,\dots,r_q,\frac{q+3}{2},q\right\}$.

The conjecture describes the possible sizes of inclusion-maximal integral point sets and is supported by the paper's computational classification. It refines the earlier expectation that only two cardinalities occur.