Mean value conjecture for the moment curve

For integers $n\geq 2$, parameters $0\leq \beta\leq n-1$, and $s\geq 1$, let $e(t)=e^{2\pi i t}$ and integrate over $[0,1]^{n-1}\times[0,N^{-\beta}]$. Mean value conjecture for the moment curve. One has

$$\int_{[0,1]^{n-1}\times [0,\frac1{N^\beta}]}\left|\sum_{k=1}^N e(kx_1+k^2x_2+\ldots+k^nx_n)\right|^{2s}dx\lesssim_\epsilon N^\epsilon\left(N^{s-\beta}+N^{2s-\frac{n(n+1)}2}\right).$$

This is a moment estimate for exponential sums along the moment curve, reflecting the expected balance between the diagonal contribution and the higher-order Vinogradov-type contribution; its resolution is not supplied in the given text.