Small cap moment-curve exponential-sum conjecture

Let $N\geq 1$, $0\leq \sigma\leq 2$, and $s\geq 1$, and write $e(t)=e^{2\pi i t}$. Then the small cap decoupling conjecture. There is a constant $C_\varepsilon$ such that

$$\int_{[0,1]^2\times[0,1/N^{\sigma}]}\left|\sum_{k=1}^N e(kx_1+k^2x_2+k^3x_3)\right|^{2s}\,dx\leq C_\varepsilon N^\varepsilon\left[N^{s-\sigma}+N^{2s-6}\right].$$

This conjecture concerns sharp $L^{2s}$ estimates for exponential sums associated with the moment curve in $\mathbb R^3$ and motivates the paper's small-cap decoupling results. The source presents it as the conjecture being proved in the case $n=3$; its resolution status is not otherwise specified here.