Local Fourier restriction conjecture

Let $S\subset\mathbb{R}^n$ be a compact $C^2$ hypersurface, possibly with boundary, with strictly positive second fundamental form. Let $E_S$ be its Fourier extension operator. Local Fourier restriction conjecture. If $p\geq\frac{2n}{n-1}$, then for every $\varepsilon>0$ and $R>1$,

$$\|E_Sf\|_{L^p(B_R)}\leq C_\varepsilon R^\varepsilon\|f\|_{L^p(d\sigma_S)}.$$

By an epsilon-removal argument, this local estimate implies the global restriction conjecture; it is not known at the endpoint in general.