Invariant-center integrals conjecture for polygon maps and vector fields

Let $f_t$ be the maps on planar polygon spaces and let $\xi$ be the vector field. The polygons evolving under these maps and under the flow of $\xi$ appear to move around invariant centers. Invariant-center integrals conjecture. For every $n$, the maps $f_t$ and the vector field $\xi$ possess two additional integrals, independent of $t$, whose values are the coordinates of these centers. The conjecture is motivated by numerical figures suggesting bounded motion around invariant centers; no proof or resolution is given.