Liouville integrability conjecture for odd-gon maps and vector fields

Let $n$ be odd, let ${\mathcal T}_n$ be the symplectic space of planar $n$-gons, let $f_t$ be the maps on this space, and let $\xi$ be the vector field. Let ${\mathcal A}$ denote the algebraic multi-area and let ${\mathcal P}$ denote the perimeter function. Liouville integrability conjecture. The maps $f_t$ and the vector field $\xi$ possess $n-1$ Poisson commuting integrals, including ${\mathcal A}$ and, in the case of $\xi$, ${\mathcal P}$. If true, the Arnold–Liouville theorem would imply quasi-periodic motion on the common level surfaces of these integrals, consistently with the experimental observations.