The extremal conjecture for $S_1$-free cyclic geometric hypergraphs

Let $\Omega_n$ be the vertices of a regular $n$-gon, and let $\operatorname{ex}_\circlearrowright(n,S_1)$ denote the maximum number of triangles in an $n$-vertex cyclic geometric hypergraph containing no copy of the configuration $S_1$. For $n\geq 5$, the conjectured extremal value is

The $S_1$ extremal conjecture.

$$\operatorname{ex}_\circlearrowright(n,S_1)=\dot{\Delta}(n)+\left\lfloor\frac n2\right\rfloor\left\lfloor\frac{n-2}{2}\right\rfloor.$$

The authors state more broadly that all extremal $S_1$-free cyclic geometric hypergraphs should belong to the previously defined family $\mathcal{H}^+(n)$. The exact value is not determined in the paper; only lower and upper bounds are proved, so this conjecture remains open.