Local restriction conjecture for the normalized extension operator

Let $\Phi:B^{n-1}(0,1)\to\mathbb{R}$ satisfy $\Phi(0)=0$, $\nabla\Phi(0)=0$, and have Hessian eigenvalues comparable to $1$. Define

$$Ef(x):=\int_{B^{n-1}(0,1)}e^{i\bar{x}\cdot\bar{\xi}}e^{ix_n\Phi(\bar{\xi})}f(\xi)\,d\xi.$$

Normalized local restriction conjecture. If $p\geq\frac{2n}{n-1}$, then for every $\varepsilon>0$ and $R>1$,

$$\|Ef\|_{L^p(B_R)}\leq C_\varepsilon R^\varepsilon\|f\|_p.$$

The paper states this as equivalent to the preceding local restriction formulation after normalizing the hypersurface and replacing surface measure by a smooth density; it remains open in general.