Bounded subcover conjecture for sunflower-free set systems

Let $k$ be an integer, and let $\cal F$ be a sunflower-free $k$-uniform set system, meaning that it contains no sunflower with the relevant number of petals.

Bounded subcover conjecture. There exist $F_1,\ldots,F_{2k}\in\cal F$ such that

$$\bigcup_{F\in\cal F}F=\bigcup_{j=1}^{2k}F_j.$$

The source presents this as a weaker version of the preceding union-size conjecture. No resolution is given in the source.