The $L^p$ square-root cancellation conjecture for nondegenerate curves in ${\mathbb R}^4$

Let $\phi_3,\phi_4$ be real analytic functions, and let $\omega_3,\omega_4$ be the objects satisfying the hypotheses of the preceding $L^{12}$ conjecture. For a finite interval $I\subset {\mathbb Z}$, define

$${\mathcal E}_{I,N}(x)=\sum_{n\in I}e(nx_1+n^2x_2+\phi_3(n/N)x_3+\phi_4(n/N)x_4).$$

Square-root cancellation in $L^p$. For $11\le p\le12$ and $\alpha\ge\beta\ge0$ satisfying $\alpha+\beta=p/2-3$, one should have

$$\int_{[0,1]\times[0,1]\times\omega_3\times\omega_4}|{\mathcal E}_{[N/2,N],N}(x)|^p\,dx\lesssim_\epsilon N^{p-3+\epsilon}.$$

This is proposed as an extension of the preceding $L^{12}$ conjecture and is motivated by the possibility of determining $L^p$ moments for curves in ${\mathbb R}^4$ in the range $8\le p\le12$. The source notes a related obstruction but does not resolve this extension.