Inverse rigidity for outer length billiards

Establish whether total integrability of the outer length billiard implies that the boundary curve is an ellipse, thereby resolving inverse rigidity for this model.

Background

The outer length billiard (also known as the fourth billiard) admits a generating function given by the sum of distances from a point outside the domain to two tangency points on the boundary.

Ellipses are shown to be totally integrable in this model, but it remains unknown whether total integrability characterizes ellipses (inverse rigidity).