Unit-ball sunflower conjecture in three dimensions

For each integer $r\geq 3$, let $V\subset\mathbb R^3$ and let $\mathcal F$ be a family of subsets of $V$, each of size $k$, such that every member of $\mathcal F$ is the intersection of $V$ with a unit ball in $\mathbb R^3$. Unit-ball sunflower conjecture. There is a constant $C=C(r)$ such that, whenever $|\mathcal F|\geq C^k$, the family $\mathcal F$ contains an $r$-sunflower.

This is a geometric special case of the sunflower conjecture for bounded-complexity set systems. The paper presents it as a natural open problem in three-dimensional Euclidean space.