Birkhoff–Poritsky conjecture for integrable billiards

Let $\Omega$ be a strictly convex domain with smooth boundary, and let its billiard map act on the cylinder parametrized by $(s,\varphi)\in\mathbb{R}/|\partial\Omega|\mathbb{Z}\times[0,\pi]$. An integrable billiard is one whose phase space contains an open set foliated by curves invariant under the billiard map. Birkhoff–Poritsky conjecture. The only integrable billiards are ellipses. This conjecture characterizes the strictly convex billiards possessing invariant-curve foliations; circles are globally integrable, while ellipses are integrable but not globally integrable. Its general validity remains open.