Saturation conjecture for the convex geometric hypergraph $M_1^{(r)}$

Let $r\ge 3$ and let $H_n^{(r)}$ be the convex geometric hypergraph defined in the source. Write $\operatorname{sat}_\circlearrowright(n,M_1^{(r)})$ for the minimum number of edges in an $M_1^{(r)}$-saturated convex geometric hypergraph on $n$ vertices. Saturation conjecture for $M_1^{(r)}$. For all $r\ge 3$ and $n$ sufficiently large in terms of $r$,

$$\operatorname{sat}_\circlearrowright(n,M_1^{(r)})=|H_n^{(r)}|.$$

Moreover, $H_n^{(r)}$ is the unique $M_1^{(r)}$-saturated convex geometric hypergraph achieving this bound provided $n$ is sufficiently large in terms of $r$. The paper proves the corresponding asymptotic lower bound for all $r$ and uniqueness when $r=3$, while the stated exact formula and uniqueness for general $r$ remain open.