Multicolor sunflower product conjecture

For integers $n$ and $k$, let ${\cal F}(n,k)$ be the collection of $k$-tuples of families ${\cal A}_i\subset 2^{[n]}$ that contain no multicolor sunflower with $k$ petals, and define

$$P(n,k):=\max_{\{{\cal A}_i\}_{i=1}^k\in{\cal F}(n,k)}\prod_{i=1}^k|{\cal A}_i|.$$

Here a multicolor sunflower with $k$ petals is a choice of sets $A_i\in{\cal A}_i$ such that all pairwise intersections equal a common core and each $A_i$ has a nonempty difference from that core. Multicolor sunflower product conjecture. For each fixed $k\geq 3$,

$$P(n,k)=\left(\frac{1}{8}+o(1)\right)2^{kn}.$$

The paper proves matching lower and upper estimates when $k=3$ only up to the constants $1/8$ and $0.13075$, respectively; the conjecture asserts that the lower bound is asymptotically sharp for every fixed $k\geq3$.