Sunflower conjecture in $(\mathbb Z/D\mathbb Z)^n$

Let $k\leq D$, and let $A\subset(\mathbb Z/D\mathbb Z)^n$. A collection of vectors is a $k$-sunflower if, in every coordinate, its entries are either all different or all equal; $A$ is $k$-sunflower-free if it contains no such collection. Sunflower conjecture in $(\mathbb Z/D\mathbb Z)^n$. There is a constant $b_k$, depending only on $k$, such that

$$|A|\leq b_k^n.$$

For $k=3$ and $D=3$, the sunflower condition is equivalent to being a three-term arithmetic progression in $\mathbb F_3^n$. The paper gives bounds for the three-sunflower case, while the stated conjecture remains open.