The generalized daisy density conjecture

For fixed integers $s$ and $t$, let $\mathcal D_r(s,t)$ consist of all $r$-sets $A$ containing a fixed $(r-t)$-set $P$ and contained in $P\cup Q$, where $Q$ is a fixed disjoint $s$-set. Let $\pi(\mathcal D_r(s,t))$ denote its limiting Turán density. The generalized daisy density conjecture.

$$\text{For fixed }s\text{ and }t,\qquad \pi(\mathcal D_r(s,t))\longrightarrow 0\quad\text{as }r\longrightarrow\infty.$$

The original daisy is the case $(s,t)=(4,2)$. The conjecture extends the paper's main asymptotic question to all fixed daisy parameters.