The daisy density conjecture

Let $[n]=\{1,\ldots,n\}$. For a set $X$, write $X^{(r)}$ for its $r$-subsets. An $r$-graph is a subset of $X^{(r)}$. Given an $(r-2)$-set $P$ and a disjoint $4$-set $Q$, the $r$-daisy $\mathcal D_r$ is the $r$-uniform hypergraph consisting of all $r$-sets $A$ satisfying $P\subset A\subset P\cup Q$. Let $\pi(\mathcal D_r)$ denote its limiting Turán density. The daisy density conjecture.

$$\pi(\mathcal D_r)\longrightarrow 0\quad\text{as }r\longrightarrow\infty.$$

This is the paper's fundamental extremal question about daisy-free uniform hypergraphs; the conjecture concerns the asymptotic density as the uniformity grows, while even the exact density for $r=3$ is presented as unknown.