Second-integral conjecture for the triangle map

Let ${\mathcal S}_3$ be the space of unit-area triangles, let $f_t:{\mathcal S}_3\to{\mathcal S}_3$ be the triangle map, and let ${\mathcal A}$ be its area integral. Second-integral conjecture. The map $f_t$ has a second integral, depending on $t$, that Poisson-commutes with the area integral ${\mathcal A}$. This is presented as a particular case of the Liouville-integrability conjecture for odd-gon maps and is intended to explain the observed phase portraits; it remains open.