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

Let ${\mathbb Z}_D=\{1,\ldots,D\}$, and call three vectors in ${\mathbb Z}_D^n$ a $3$-sunflower when their entries in each coordinate are either all equal or all distinct. Alon–Shpilka–Umans' weak sunflower conjecture. For the fixed integer $D$ considered here, there exist $\varepsilon>0$ and $n_0$ such that for every $n\geq n_0$, 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 the fixed-$D$ variant of the preceding sunflower conjecture and is presented as a question about whether the assertion holds for small $D$.