Non-power cardinality conjecture for modular $p$-free sets

Let $p$ be an odd prime. A set is modular $p$-free if it is modular and contains no nontrivial arithmetic progression of length $p$; $|A|$ denotes its cardinality.

Non-power cardinality conjecture. For every $n\in\mathbb{N}\cup\{0\}$, there exists a modular $p$-free set $A$ such that

$$|A|\ne (p-1)^n.$$

The paper states that the only resolved case is $p=3$, with examples supplied there; the general assertion is therefore open.