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Sequence of rotational invariant curves with rotation numbers tending to 1/2

Prove that if a Birkhoff billiard admits a sequence of rotational invariant curves whose rotation numbers tend to 1/2, then the boundary curve is an ellipse.

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Background

Innami proved that if there exists a sequence of convex caustics with rotation numbers approaching 1/2, then the billiard table is an ellipse.

The authors ask whether the same conclusion holds when one assumes only rotational invariant curves (not necessarily convex caustics) with rotation numbers tending to 1/2.

References

In this section we formulate natural open questions related to the results discussed in previous sections. (3) Suppose for the Birkhoff billiard in $\gamma$ there exist a sequence $\alpha_n\subset\mathbb A$ of rotational invariant curves with rotation numbers tending to $\frac 12$. Is it true that in this case $\gamma$ is an ellipse? If all invariant curves $\alpha_n$ correspond to convex caustics, then this is true .

— Integrable Billiards and Related Topics (2510.03790 - Bialy et al., 4 Oct 2025) in Section 9 (Open questions), Subsection Birkhoff billiards, item (3)