Novikov–Dynnikov generic TCI-stability conjecture

Let $\varepsilon_c$ be a level hypersurface of a smooth quasiperiodic function on ${\mathbb T}^4$, and let $\Pi\in G_{4,2}(\mathbb R)$ be a $2$-plane. The level is topologically completely integrable (TCI) for $\Pi$ when all $\Pi$-sections are either compact or have a strong asymptotic direction; it is TCI stable when this persists under small perturbations of $\Pi$ and the function. Novikov–Dynnikov's conjecture. A generic level $\varepsilon_c$ is TCI stable for almost all $\Pi\in G_{4,2}(\mathbb R)$. The surrounding theorem establishes an open dense set of functions and an open dense set of planes for which every level is TCI, with bounded cylinders for open sections, but the stronger almost-everywhere stability assertion is stated separately and remains open in the source.