The Main Conjecture for discrete Fourier restriction mean values

Let $k\geqslant 1$, let $s$ be a positive integer, let $N$ be a positive integer, and let $({\mathfrak a}_n)$ be any complex sequence. Writing $f_{\boldsymbol {\mathfrak a}}({\boldsymbol \alpha};N)$ for the associated exponential sum and integrating over the relevant $k$-dimensional torus, the conjecture is that, for every $\varepsilon>0$,

$$\oint |f_{\boldsymbol {\mathfrak a}} ({\boldsymbol \alpha};N)|^{2s}{\,{\rm d}}{\boldsymbol \alpha} \ll N^\varepsilon \left(1+N^{s-k(k+1)/2}\right) \biggl( \sum_{|n|\leqslant N}|{\mathfrak a}_n|^2\biggr)^s.$$

Main Conjecture. For every $\varepsilon>0$ and every complex sequence $({\mathfrak a}_n)$, the displayed bound holds.