Smoothness conjecture for the Poincaré problem on convex simple domains

Let $f(x)$) be a smooth forcing function, let $r(\lambda)$ be a sufficiently irrational rotation number, and let the Poincaré problem act on a convex $\lambda$-simple domain, in the sense of Dyatlov et al.

Smoothness conjecture. The solution $u(t)$ of the Poincaré problem is smooth.

The conjecture proposes extending the smoothness result established in the paper for the square to other convex $\lambda$-simple domains. Its general validity is left as a future direction.