Alon–Shpilka–Umans' sunflower conjecture over ${\mathbb Z}_D^n$

Let ${\mathbb Z}_D=\{1,\ldots,D\}$. Vectors $v_1,v_2,v_3\in{\mathbb Z}_D^n$ form a $3$-sunflower if, in every coordinate, their entries are either all equal or all distinct. Alon–Shpilka–Umans' sunflower conjecture. There exist $\varepsilon>0$, $D_0$, and $n_0$ such that for every $D\geq D_0$ and $n\geq n_0$, every collection ${\cal F}$ of vectors in ${\mathbb Z}_D^n$ with

$$|{\cal F}|\geq D^{(1-\varepsilon)n}$$

contains a $3$-sunflower. This is an equivalent vector formulation of the Boolean-cube sunflower conjecture and is widely believed to be true, but remains open.