Tabachnikov's projective Birkhoff conjecture

Let $C$ be a strictly convex closed curve equipped with a projective billiard structure, and call the billiard Birkhoff integrable when its boundary and closed caustics form a foliation. Tabachnikov's projective Birkhoff conjecture. In every Birkhoff-integrable projective billiard, the boundary and the closed caustics forming the foliation are ellipses whose projective-dual conics form a pencil. This is a projective generalization of Birkhoff's conjecture. The paper proves the assertion for $C^4$-smooth germs carrying a rationally integrable dual billiard structure, but the full conjecture remains open.