Tabachnikov's dual Birkhoff conjecture for projective billiards

Let $\gamma\subset\mathbb{RP}^2$ be a strictly convex closed curve. A dual billiard structure on $\gamma$ is a family of non-trivial projective involutions on the projective tangent lines, fixing their tangency points. Call the dual billiard integrable when there is a $C^0$-foliation by closed strictly convex invariant curves on a neighborhood of $\gamma$ on its concave side, with $\gamma$ as a leaf. Tabachnikov's dual Birkhoff conjecture. For every integrable dual billiard, the underlying curve and the corresponding invariant curves forming the foliation are conics forming a pencil. The conjecture is dual to the projective Birkhoff conjecture and is proved in the paper under the additional assumption of rational integrability and $C^4$ smoothness; it remains open in this general form.