Smooth extension conjecture for integrable billiards

Let $\Omega$ be a smooth integrable billiard domain, and let $\mathcal E$ be an ellipse with the same perimeter and the same Lazutkin perimeter. Consider the conjugacy between the billiard near-boundary dynamics of $\Omega$ and $\mathcal E$ described in the source. Smooth extension conjecture. This conjugacy can be smoothly extended, in Whitney's sense, up to the boundary. This conjecture concerns the extension problem for conjugacies of integrable billiards and is presented as closely related to Birkhoff's conjecture; its status is not resolved in the source.