Novikov and Dynnikov's generic TCI-stability conjecture in the four-torus

Let $\varepsilon:\mathbb{T}^4\to\mathbb{R}$ be a smooth function, let $\varepsilon_c=\varepsilon^{-1}(c)$ be a level hypersurface, and let $\Pi\in G_{4,2}(\mathbb{R})$ be a $2$-plane. The level is topologically completely integrable (TCI) for $\Pi$ if all $\Pi$-sections are either compact or have a strong asymptotic direction, and it is TCI stable if this remains true after small perturbations of $\Pi$ and $\varepsilon$. Novikov and Dynnikov's conjecture. A generic level $\varepsilon_c$ is TCI stable for almost all $\Pi\in G_{4,2}(\mathbb{R})$. The preceding theorem gives TCI for an open dense set of planes but not stability for almost all planes, so the stronger stability assertion remains open in the source.