Erdős–Szemerédi's sunflower conjecture in the Boolean cube

A $3$-sunflower is a collection of three sets having the same pairwise intersections, and $[n]=\{1,\ldots,n\}$. Erdős–Szemerédi's Boolean-cube sunflower conjecture. There exists an $\varepsilon>0$ such that for every $n\geq 2$, every collection ${\cal F}$ of subsets of $[n]$ with

$$|{\cal F}|\geq 2^{(1-\varepsilon)n}$$

contains a $3$-sunflower. This is a special case implied by the classical sunflower conjecture and is itself still open.