Local Stein restriction conjecture for positively curved surfaces in three dimensions

Let $S$ be a smooth compact surface in $\mathbb{R}^3$ with positive definite second fundamental form, and let $d\sigma$ be its induced Lebesgue measure. Define the Fourier extension operator by

$$\mathcal E_Sg(x):=\int_S g(\omega)e^{2\pi i x\cdot\omega}\,d\sigma.$$

For $R\geq 1$, let $B_R$ be the ball centred at the origin with radius $R$ in $\mathbb{R}^3$. Local Stein restriction conjecture. For every $\varepsilon>0$ and every $p>3$, there is a constant $C(\varepsilon)$ such that

$$\|\mathcal E_Sg\|_{L^p(B_R)}\leq C(\varepsilon)R^\varepsilon\|g\|_{L^\infty(S,d\sigma)}.$$

This is the local version obtained from the global restriction estimate by the standard epsilon-removal framework. The surrounding discussion presents the restriction problem as broadly open, so this local formulation remains open as well.