The discrete restriction conjecture for toral eigenfunctions

Let $p\geq 2$ and $d\geq 2$. Let $\mathbb{T}^d=\mathbb{R}^d/\mathbb{Z}^d$ be the square torus, and let $e_N$ be an eigenfunction of $\sqrt{-\Delta}$ with eigenvalue $2\pi N$. The discrete restriction conjecture. One should have

$$\|e_N\|_{L^p(\mathbb{T}^d)}\lesssim_{\varepsilon}N^{\varepsilon}\left(N^{\frac{d-2}{2}-\frac{d}{p}}+1\right)\|e_N\|_{L^2(\mathbb{T}^d)}.$$

Moreover, the extra factor $N^{\varepsilon}$ can be omitted if $d\geq 5$. This conjecture gives the expected sharp numerology for restriction estimates on lattice points of spheres. The paper proves the bound without the loss when $d\geq 5$ and $p>\frac{2d}{d-4}$, while in a wider range it obtains a logarithmic loss; the full stated range remains open.