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Minkowski billiards: integrability characterizes Euclidean norms?

Prove or disprove that the Minkowski billiard inside the unit ball of a norm N is totally integrable if and only if N is the Euclidean norm.

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Background

Minkowski (Finsler) billiards generalize Birkhoff billiards by replacing the Euclidean norm with a (possibly asymmetric) norm and using its unit ball as the billiard domain.

The authors ask whether integrability within the unit ball rigidly forces the norm to be Euclidean, a question communicated by Y. Ostrover, with links to symplectic and convex geometry.

References

In this section we formulate natural open questions related to the results discussed in previous sections. (1) Prove or disprove that a Minkowski billiard corresponding to the norm $N$ inside the unit ball of $N$ is totally integrable if and only if $N$ is a Euclidean norm (communicated by Y.Ostrover).

— Integrable Billiards and Related Topics (2510.03790 - Bialy et al., 4 Oct 2025) in Section 9 (Open questions), Subsection Minkowski billiards, item (1)