The general multilinear Fourier extension conjecture for the paraboloid

Let $k\geq2$, and let $U_1,\ldots,U_k$ parametrize transversal caps of the paraboloid $x\mapsto|x|^2$ in $\mathbb{R}^{d+1}$. For functions $g_j$ on $U_j$, write $p'$ for the Hölder conjugate of $p$. The general multilinear extension conjecture. If

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

then

$$\left\|\prod_{j=1}^{k}\mathcal{E}_{U_j}g_j\right\|_{L^{q/k}(\mathbb{R}^{d+1})}\lesssim_{p,q}\prod_{j=1}^{k}\|g_j\|_{L^p(U_j)}.$$

This extends the $L^2$ multilinear conjecture to $L^p$ inputs and is intended to clarify the role of transversality; the supplied text gives no resolution status.