Birkhoff–Poritsky conjecture for Kepler billiards

Determine whether, among bounded strictly convex planar domains with mathcal{C}^2 boundary, the only Kepler billiards integrable at all energy levels are ellipses with the centre of attraction located at one of the foci.

Background

A Kepler billiard is a mechanical billiard in which the particle moves under a Newtonian central attraction and reflects elastically at the boundary. Ellipses with the centre of attraction at a focus provide integrable examples at every energy level.

The cited work has partially addressed this conjecture, but the paper notes an unresolved exceptional possibility involving a special position of the attraction centre. The conjecture is therefore explicitly presented as an open classification problem.