Paraboloid exponential-sum restriction conjecture

Let $d\geq 2$, $N\geq 1$, and define

$$\mathbb{P}^{d-1}_N=\{\mathbf n=(n_1,\ldots,n_{d-1},n_1^2+\cdots+n_{d-1}^2)\in\mathbb{Z}^d:1\leq n_i\leq N\},$$ $$S^{\mathbb{P}}_{a,d}(x,N)=\sum_{\mathbf n\in\mathbb{P}^{d-1}_N}a_{\mathbf n}e(\mathbf n\cdot x).$$

Let $\mathcal{M}$ be a real analytic hypersurface in $\mathbb{T}^d$ with nonnegative curvature and surface measure $\sigma$. Paraboloid restriction conjecture. For every $d\geq 2$ and $N\geq 1$,

$$\|S^{\mathbb{P}}_{a,d}(x,N)\|_{L^2(d\sigma)}\lesssim\|a\|_{\ell^2}.$$

The source calls this conjecture tentative; it is motivated by the positive curvature of the paraboloid and its connection with the free Schrödinger equation on the torus, and remains open.