Green's Open Problem 39: Variant Theta

Similar questions are interesting with $\sqrt{p}$ replaced by $p^\theta$ for any $\theta \le 1/2$. [Gr24]

NOTE: using $C p^\theta$ translates as stated makes the conjecture trivially false by the pigeonhole principle. Indeed for a set of size $p^\theta$, we cover at most $C p^{2\theta}$ elements, which is strictly less than $p$ for $\theta < 1/2$. We interpret the question as asking whether $O(p^{1-\theta})$ translates suffice. This generalizes the main conjecture where $\sqrt{p} = p^{1-1/2}$.