Ball's weak cylinder conjecture for point sets in finite affine 3-space

Let $S\subseteq\mathbb{F}_p^3$ be a set of $p^2$ points, where $p$ is prime. The cylinder condition means that $S$ is the union of $p$ pairwise disjoint parallel lines, and hence has size $p^2$. The weak cylinder conjecture. If the number of non-determined directions of $S$ satisfies $|N_S|\geq p$, then $S$ is a cylinder. This is one of Ball's proposed forms of the cylinder conjecture over prime fields. The problem concerns the structure of sets with few determined directions in higher-dimensional finite affine spaces and is presented in the source as an open question.