Unbounded integral point sets over prime residue rings

For a prime $p$, let $\dot{\mathcal{I}}(\mathbb{Z}_p,2)$ denote the maximum cardinality of an integral point set in the plane over $\mathbb{Z}_p$ in general position. Unboundedness conjecture. For each positive integer $l$ there is a prime threshold $p'$ such that for all primes $p\ge p'$,

$$\dot{\mathcal{I}}(\mathbb{Z}_p,2)\ge l.$$

The surrounding discussion notes that several non-isomorphic sets attain the upper bound and suggests only the weaker observation that the value is at least $7$ for sufficiently large primes. Over the integers, the corresponding question for seven points is stated to remain unsolved.