Nodal-line width conjecture for eigenfunctions of the two-dimensional torus

Let $\mathbb T^2$ be the two-dimensional torus, let $\varphi$ be an eigenfunction with eigenvalue $E=\lambda^2$, and let $C\subset\mathcal N_\varphi$ be a regular arc. For a convex curve, write $w(C)$ for the minimal distance between a pair of parallel supporting lines. Nodal-line width conjecture. For every $\varepsilon>0$, there is $c_\varepsilon>0$ such that

$$w(C)<c_\varepsilon\lambda^{-1+\varepsilon}.$$

This is proposed as a substitute for the curvature behavior of nodal lines of random plane waves. It is consistent with numerical evidence, and the paper proves the weaker general bound $w(C)<C_\varepsilon\lambda^{-1/3+\varepsilon}$, as well as a stronger result for most of the nodal line.