The Novikov conjecture on exceptional directions

Let $F\to oalign{\vskip\baselineskip}$ be a generic smooth function on the three-torus, and let ${\cal E}_F$ denote the exceptional set of directions associated with its quasiperiodic level sets. Novikov's conjecture. For a generic $F\in C^\infty({\mathbb T}^3)$, the set ${\cal E}_F$ has zero measure and Hausdorff dimension strictly between $1$ and $2$. This concerns the expected fractal geometry of the exceptional directions, complementing the stability-zone description of open level-set components; the source leaves the claim unanswered.