Discrete restriction conjecture for the curve $(x,x^3)$

Let $N\in\mathbb{N}$ and let $a\in\ell^2(\mathbb{Z})$. Define the truncated extension operator

$$Ea(\alpha,\beta):=\sum_{|n|\leq N}a(n)e(\alpha n^3+\beta n),$$

where $(\alpha,\beta)\in\mathbb{T}^2$ and $e(t)=e^{2\pi i t}$. Discrete restriction conjecture. For each $p\in[1,\infty]$, there exists a constant $C_p>0$ such that, for all $N\in\mathbb{N}$ and all $a\in\ell^2(\mathbb{Z})$,

$$\|Ea\|_{L^p(\mathbb{T}^2)}\leq C_p\left(1+N^{\frac12-\frac4p}\right)\|a\|_{\ell^2(\mathbb{Z})}.$$

This is the expected sharp discrete restriction estimate for the cubic moment curve, motivated by circle-method heuristics; its resolution is not indicated in the supplied text.