The strengthened multilinear restriction conjecture

Let $S_1,\ldots,S_d$ be transversal smooth codimension-one submanifolds of $\mathbb{R}^d$, with induced measures $d\sigma_j$. Let $p'$ be the Hölder conjugate of $p$. Strengthened multilinear restriction conjecture. If $\frac{1}{q}\leq\frac{d-1}{2d}$ and $\frac{1}{q}\leq\frac{d-1}{d}\frac{1}{p'}$, then

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

The source presents this as a stronger form suggested by standard examples in the special case of full $d$-linearity.