Sharp discrete restriction conjecture for two polynomial phases

Let $\phi_1,\phi_2\in\mathbb{Z}[n]$ be polynomials with respective degrees $k_1$ and $k_2$, and define

$$Ea(\alpha_1,\alpha_2):=\sum_{|n|\leq N}a(n)e\bigl(\alpha_1\phi_1(n)+\alpha_2\phi_2(n)\bigr),$$

where $a\in\ell^2(\mathbb{Z})$ and $(\alpha_1,\alpha_2)\in\mathbb{T}^2$. Assume $\max\{k_1,k_2\}\geq3$ and that $\phi_1'$ and $\phi_2'$ are linearly independent over $\mathbb{Q}$. Sharp two-polynomial discrete restriction conjecture. For each $p\in[1,\infty]$,

$$\|Ea\|_{L^p(\mathbb{T}^2)}\lesssim\left(1+N^{\frac12-\frac{k_1+k_2}{p}}\right)\|a\|_{\ell^2(\mathbb{Z})}$$

as $N\to\infty$. This is presented as the expected sharp bound in general; related estimates are known in special cases, but the stated general assertion remains unresolved in the supplied text.