Regularity of maximizers of the weighted Cheeger functional on polygons

Let $\mathcal{J}$ be the functional on the class $\mathcal{P}_n$ of simple polygons used in the paper. For $n\in\{3,4\}$, the maximizers of $\mathcal{J}$ over $\mathcal{P}_n$ are regular and inscribed in a circle centered at the origin. Polygonal Cheeger maximizer conjecture. The result stated above holds for all $n\geq 3$. The proof is given only for triangles and quadrilaterals and does not extend to $n\geq 5$; the conjecture concerns the structure of all optimal polygons.