Hooke billiard classification conjecture for smooth connected walls

A Hooke billiard is a mechanical billiard in $\mathbb{R}^2$ with Hooke potential $U=f r^2$, where $r$ is the distance from a fixed center. Let its reflection wall be smooth and connected, and require integrability on all regular energy hypersurfaces. Hooke billiard classification conjecture. The only such Hooke billiards are those whose reflection wall is a branch of a centered conic section or a line. This is proposed as a classification analogue of the Birkhoff–Poritsky conjecture; the stated classification remains open.