Finite-subset saturation correspondence conjecture

For $P\subseteq\mathbb{R}^2$, let $M_P$ denote the associated finite forbidden $0$–$1$ matrix when $P$ is finite, and let $\operatorname{osat}(n,P)$ be the continuous saturation function defined using saturating open subsets of $[0,n]^2$. Finite-subset saturation correspondence conjecture. For every finite subset $P\subseteq\mathbb{R}^2$, $\operatorname{osat}(n,P)$ is defined and

$$\operatorname{osat}(n,P)=\Theta\bigl(\operatorname{sat}(n,M_P)\bigr).$$

This would establish a continuous counterpart of the relationship between continuous Turán numbers and forbidden-matrix extremal functions; both the existence assertion and the asymptotic correspondence are open in the source.