The multilinear restriction conjecture for curved hypersurfaces

Let $S_i\subset\mathbb{R}^n$, $i=1,\ldots,k$, be smooth, compact hypersurfaces, where $1\leq k\leq n$. For functions $f_i$ supported in neighborhoods $U_i$ of $S_i$, write $\mathcal R^*(2\times\cdots\times 2\rightarrow p)$ for the estimate

$$\left\|\prod_{i=1}^k\mathcal E_i f_i\right\|_{L^p(\mathbb{R}^n)}\lesssim\prod_{i=1}^k\|f_i\|_{L^2(U_i)}.$$

Assume the standard transversality condition: there is $\nu>0$ such that

$$\left|N_1(\zeta_1)\wedge\cdots\wedge N_k(\zeta_k)\right|\geq\nu$$

for every $\zeta_i\in S_i$, where $N_i(\zeta_i)$ is the unit normal to $S_i$ at $\zeta_i$. Multilinear restriction conjecture for curved hypersurfaces. Under appropriate transversality and curvature conditions on the surfaces $S_i$, the estimate $\mathcal R^*(2\times\cdots\times 2\rightarrow p)$ holds for every

$$p\geq p(k)=\frac{2(n+k)}{k(n+k-2)}.$$

This is the non-generic multilinear restriction problem, where curvature is expected to improve on the exponent $2/(k-1)$ suggested by transversality alone. The source states the conjectured bound but does not specify the precise curvature hypotheses, and its resolution status is not established here.