Centrally-symmetric rigidity for outer billiards (analogue of Bialy–Mironov)

Determine whether a centrally symmetric outer billiard analogue of the Bialy–Mironov rigidity holds; specifically, establish that for a centrally symmetric C^2 convex curve whose outer billiard has an invariant curve of 4-periodic orbits and is totally integrable on the associated region of phase space, the boundary must be an ellipse.

Background

For outer billiards, the authors proved a global rigidity result: total integrability on the entire exterior phase space implies the boundary is an ellipse.

They note a plausible centrally symmetric partial-phase-space analogue, modeled on their Birkhoff result with a 4-periodic invariant curve, but currently no proof is known.