Orbital maximax is bounded below by constant fraction of number of ordered pairs of distinct elements for groups of fixed prime power order

Statement
Suppose $$p$$ is a (fixed) prime number. Consider the fact about::orbital maximax problem for finite $$p$$-groups acting on a set $$S$$ of size $$n$$: we want to find the maximum possible size of the largest orbital under the action of a finite $$p$$-group on $$S$$.

The claim is that there is a constant $$c_0$$ (depending on $$p$$, and in fact of the order of $$p^2$$), such that for any $$n$$, there is a group action such that size of the largest orbital is $$\ge n(n-1)/c_0$$.