Hickman–Wright's Kakeya set conjecture over finite cyclic rings

Let $N,n\in\mathbb{N}$, and let $S\subseteq (\mathbb{Z}/N\mathbb{Z})^n$ be a Kakeya set, meaning that $S$ contains a line in every direction in $\mathbb{P}(\mathbb{Z}/N\mathbb{Z})^{n-1}$. Hickman–Wright's Kakeya set conjecture. For all $\epsilon>0$ and $n\in\mathbb{N}$ there exists a constant $C_{n,\epsilon}$ such that every Kakeya set $S\subset (\mathbb{Z}/N\mathbb{Z})^n$ satisfies

$$|S|\geq C_{n,\epsilon}N^{n-\epsilon}.$$

The conjecture gives a near-full-dimensional lower bound for Kakeya sets over $\mathbb{Z}/N\mathbb{Z}$, uniformly for general $N$. The source states that this conjecture is resolved in the paper, so its status is solved.