Linear-union 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 specified number of petals under consideration.

Linear-union conjecture. There exists a constant $D>0$ such that

$$\left|\bigcup_{F\in\cal F}F\right|\leq Dk^2.$$

The source states that this conjecture would imply an unconditional strong upper bound for the size of every sunflower-free $k$-uniform set system. No resolution is given in the source.