Novikov–Maltsev Hausdorff-dimension conjecture for chaotic sections

Let $M$ be a fixed surface, and let $H$ be a covector whose corresponding sections of the level surfaces associated with $M$ may exhibit chaotic behavior. Novikov–Maltsev conjecture. The Hausdorff dimension of the set of covectors $H$ admitting chaotic sections is smaller than $2$. This is a strongest form of the preceding measure-zero questions concerning chaotic sections; the source does not indicate whether it has been proved.