Sharp discrete restriction conjecture for a univariate polynomial

Let $\phi\in\mathbb{Z}[n]$ be a polynomial of degree $k\geq3$, and let $a\in\ell^2(\mathbb{Z})$. Define the extension operator

$$Ea(\alpha):=\sum_{|n|\leq N}a(n)e\bigl(\alpha\phi(n)\bigr),$$

where $\alpha\in\mathbb{T}$ and $e(t)=e^{2\pi i t}$. Sharp univariate discrete restriction conjecture. For each $p\in[1,\infty]$,

$$\|Ea\|_{L^p(\mathbb{T}^2)}\lesssim\left(1+N^{\frac12-\frac{k}{p}}\right)\|a\|_{\ell^2(\mathbb{Z})},$$

as $N\to\infty$. The claim is stated as the expected sharp bound in general; the surrounding text gives sharp results for some cubic exponents and discusses improvements for special high-degree monomials, but does not establish the conjecture in full.