The 3-daisy density conjecture

Let $\mathcal D_3$ be the daisy formed from a fixed $1$-set and a disjoint $4$-set, and let $\pi(\mathcal D_3)$ be its limiting Turán density. The 3-daisy density conjecture.

$$\pi(\mathcal D_3)=\frac12.$$

The conjecture is motivated by iterating blow-ups of the complement of the Fano plane, which yields density $1/2$; the source states that the exact limiting density was not known.