Ball's strong cylinder conjecture for p-divisible 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 set $S$ is $p$-divisible if every affine hyperplane in $\mathbb{F}_p^3$ intersects $S$ in a number of points congruent to $0\pmod p$. A cylinder is the union of $p$ pairwise disjoint parallel lines. The strong cylinder conjecture. If $S$ is $p$-divisible, then $S$ is a cylinder. The strong form replaces the directional hypothesis of the weak form by the algebraic condition of $p$-divisibility; the source presents it as an open problem.