Bollobás–Leader–Malvenuto conjecture on Turán densities of daisies

For integers $r \geq 3$ and $t \geq 2$, an $r$-uniform $t$-daisy $\mathcal D^t_r$ is the collection of sets

$$\{S \cup T: T\subset U,\ |T|=t\}$$

where $|S|=r-t$, $|U|=2t$, and $S\cap U=\emptyset$. Its Turán density is

$$\pi(\mathcal D^t_r)=\lim_{n\to\infty}\frac{\operatorname{ex}(n,\mathcal D^t_r)}{\binom nr},$$

where $\operatorname{ex}(n,\mathcal D^t_r)$ is the maximum size of a family of $r$-element subsets of $\{1,2,\ldots,n\}$ containing no copy of $\mathcal D^t_r$.

Bollobás–Leader–Malvenuto conjecture. For all $t\geq 2$,

$$\lim_{r\to\infty}\pi(\mathcal D^t_r)=0.$$

This conjecture was made by Bollobás, Leader and Malvenuto, and independently by Bukh. It remains open even for $t=2$; for comparison, the first unknown case is $\pi(\mathcal D^2_3)$, for which the known lower bound is $1/2$.