Novikov and Dynnikov's Hausdorff-dimension conjecture for the exceptional set

Let $f:\mathbb{T}^3\to\mathbb{R}$ be a smooth function and let $\mathcal{E}(f)$ be its exceptional set of directions. Let $\dim_H$ denote Hausdorff dimension. Novikov and Dynnikov's conjecture. Whenever $\mathcal{E}(f)$ is nonempty, its Hausdorff dimension is strictly between $1$ and $2$ for every $f$. The source describes the zero-measure question as unresolved and presents this as a stronger conjecture; it gives no resolution of the dimension claim.