Bourgain–Rudnick conjecture on nodal intersections with curves

Let ${\mathbf{T}}^2={\mathbf{R}}^2/{\mathbf{Z}}^2$ be the two-dimensional flat torus, let $F$ be a real-valued Laplacian eigenfunction with eigenvalue $\lambda^2$, and let ${\mathcal C}\subset {\mathbf{T}}^2$ be a unit-length real-analytic curve with parametrization $\gamma$ satisfying

$$\|\gamma'(t)\|_2=1\quad\text{and}\quad\|\gamma”(t)\|_2>c$$

for some positive constant $c$. Define the nodal intersection count by

$${\mathcal Z}(F):=\#\{x\in{\mathcal C}:F(x)=0\}.$$

Bourgain–Rudnick conjecture. The lower bound for the number of nodal intersections is of order $\lambda$; equivalently,

$${\mathcal Z}(F)\gg\lambda.$$

This would improve the known uniform lower bound ${\mathcal Z}(F)\gg\lambda^{1-\varepsilon}$, while the matching upper bound ${\mathcal Z}(F)\ll\lambda$ is known under the stated assumptions on ${\mathcal C}$.