Novikov's exceptional-set dimension conjecture

Let $f:{\mathbb T}^3\to\mathbb R$ be a function, and let ${\cal E}(f)$ be its exceptional set of directions, namely the directions not belonging to the stability-zone set. Novikov's conjecture. Whenever ${\cal E}(f)$ is non-empty, the Hausdorff dimension of ${\cal E}(f)$ is strictly between $1$ and $2$ for every $f$. The source says that zero measure is plausible but unknown and presents this as a stronger conjecture; it remains unresolved in the text.