Multilinear Fourier extension conjecture for transversal curved hypersurfaces

Let $k\geq 2$. Let $S_1,\ldots,S_k\subseteq\mathbb{R}^{d}$ be smooth hypersurfaces equipped with measures $\sigma_1,\ldots,\sigma_k$, respectively. They are transversal if there is some $c>0$ such that

$$|v_1\wedge\cdots\wedge v_k|\geq c$$

for every choice of unit normal vectors $v_j$ to $S_j$. Let $p'$ denote the Hölder conjugate of $p$, and let $\mathcal{R}^{\ast}_{\sigma_1,\ldots,\sigma_k}(p\times\cdots\times p\rightarrow q/k)$ mean that

$$\left\|\prod_{j=1}^{k}\widehat{g_j\mathrm{d}\sigma_j}\right\|_{L^{q/k}(\mathbb{R}^{d})}\lesssim\prod_{j=1}^{k}\|g_j\|_{L^p(\mathrm{d}\sigma_j)}$$

for all $g_j\in L^p(\sigma_j)$. Multilinear Fourier extension conjecture. If the hypersurfaces have everywhere positive principal curvatures and

$$\frac{1}{q}<\frac{d-1}{2d},\qquad \frac{1}{q}\leq\frac{d+k-2}{d+k}\frac{1}{p'},\qquad \frac{1}{q}\leq\frac{d-k}{d+k}\frac{1}{p'}+\frac{k-1}{k+d},$$

then $\mathcal{R}^{\ast}_{\sigma_1,\ldots,\sigma_k}(p\times\cdots\times p\rightarrow q/k)$. This conjecture predicts the range of multilinear extension estimates produced by transversality and curvature; the source does not provide evidence resolving the stated range.