Dichotomy conjecture for continuous saturation functions

For $P\subseteq\mathbb{R}^2$, call an open set $S\subseteq[0,n]^2$ saturating for $P$ if $S$ avoids $P$ and every proper open superset of $S$ in $[0,n]^2$ contains $P$. Define $\operatorname{osat}(n,P)$ as the infimum of the area $\mu_2(S)$ over such saturating open sets, when one exists. Continuous saturation dichotomy conjecture. For every $P\subseteq\mathbb{R}^2$ for which $\operatorname{osat}(n,P)$ is defined, either

$$\operatorname{osat}(n,P)=O(1)\qquad\text{or}\qquad\operatorname{osat}(n,P)=\Theta(n),$$

with constants depending on $P$. This is modeled on the corresponding dichotomy for $0$–$1$ matrix saturation functions and is open for continuous forbidden subsets.