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Deforming a planar ellipse in R^4 while preserving wire-billiard integrability

Determine whether one can deform a planar ellipse embedded in R^4 into a nonplanar closed curve such that the corresponding wire billiard remains integrable.

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Background

In R4 there exist totally integrable wire billiards associated with certain closed curves (e.g., toric knots on S3).

The authors ask if a planar ellipse can be deformed within R4 so that the wire billiard remains integrable along the deformation.

References

In this section we formulate natural open questions related to the results discussed in previous sections. (2) Can one deform in $\mathbb R4$ an ellipse $E\subset\mathbb R2\subset \mathbb R4$ so that the wire billiard remains integrable?

— Integrable Billiards and Related Topics (2510.03790 - Bialy et al., 4 Oct 2025) in Section 9 (Open questions), Subsection Wire and cone billiards, item (2)