Asymptotic saturation conjecture for the convex geometric hypergraph $S_3$

Let $S_3$ be the two-edge convex geometric $3$-uniform hypergraph denoted by $S_3$ in the source, and let $\operatorname{sat}_\circlearrowright(n,S_3)$ denote its saturation number on $n$ vertices. Saturation conjecture for $S_3$.

$$\operatorname{sat}_\circlearrowright(n,S_3)=\Theta(n\log_2 n).$$

The paper determines the order of magnitude of the saturation number for every two-edge convex geometric $3$-uniform hypergraph except $S_3$; the displayed estimate is presented as the authors' best guess and remains open.