Uniform restriction conjecture for toral eigenfunctions

Let $d\geq 2$ and let $\Sigma\subset\mathbb T^d$ be a real analytic hypersurface. For an eigenfunction $\varphi_\lambda$ of the Laplacian on $\mathbb T^d$, write $\|\varphi_\lambda\|_2$ for its ambient $L^2$ norm and $\|\varphi_\lambda\|_{L^2(\Sigma)}$ for its restriction norm. Uniform restriction conjecture. There is a constant $C_\Sigma$ such that all eigenfunctions satisfy

$$\|\varphi_\lambda\|_{L^2(\Sigma)}\leq C_\Sigma\|\varphi_\lambda\|_2.$$

If moreover $\Sigma$ has nowhere vanishing curvature and $\lambda>\lambda_\Sigma$, for some $c_\Sigma>0$ one also has

$$\|\varphi_\lambda\|_{L^2(\Sigma)}\geq c_\Sigma\|\varphi_\lambda\|_2.$$

The statement would extend the main restriction theorem from the cases treated in the paper to arbitrary dimensions. The supplied text says that this extension remains unsettled, so the conjecture is open.