The global cone conjecture for moduli spaces of simple polygons

Let $S(n)$ be the space of positively oriented simple labeled $n$-gons modulo oriented similarity, and let $D_n$ be its dihedral symmetry group. Define the moduli space of simple polygons by ${\mathcal M}(n)=S(n)/D_n$. Let $C(X)=\bigl(X\times(0,1]\bigr)/(X\times\{1\})$ be the open cone over a topological space $X$. For the regular $n$-gon, let $\Sigma$ and $T$ act on $\mathbb{S}^{2n-5}$ by the linearized dihedral action, with

$$(z_1,z_2,\dots,z_{n-2})\overset{\Sigma}\longrightarrow(e^{4\pi i/n}z_1,e^{6\pi i/n}z_2,\dots,e^{2\pi i(n-1)/n}z_{n-2}),$$ $$(z_1,z_2,\dots,z_{n-2})\overset{T}\longrightarrow-(e^{-4\pi i/n}\overline{z}_1,e^{-6\pi i/n}\overline{z}_2,\dots,e^{-2\pi i(n-1)/n}\overline{z}_{n-2}).$$

Global cone conjecture for moduli spaces of simple polygons. ${\mathcal M}(n)$ is homeomorphic to $C(\mathbb{S}^{2n-5}/\langle\Sigma,T\rangle)$. A local version holds in a neighborhood of the regular $n$-gon; the conjecture asserts that this cone description extends globally, and the existence of the corresponding global regularizing flow remains open according to the authors.