Monochromatic sunflower conjecture

Let $r\geq 1$ and let $X$ be a finite ground set. Let $\mathcal{F}$ be a $w$-set system, meaning that every member has size at most $w$. A random $\delta$-fraction coloring colors each element of $X$ red independently with probability $\delta$, leaving the other elements uncolored. An $r$-sunflower is a collection of $r$ sets whose pairwise intersections equal their common intersection; its petals are the sets after removing that common intersection. Monochromatic sunflower conjecture. For every $\delta>0$, there exists a constant $K=K(r,\delta)$ such that if $|\mathcal{F}|\geq K^w$, then with high probability the family contains an $r$-sunflower whose petals are all red.

This conjecture is a monochromatic strengthening of the sunflower phenomenon. The source presents it as another variant related to the rainbow sunflower conjecture and does not state a resolution of the $K^w$ bound.