The smooth Alpert trilinear Fourier extension conjecture for the paraboloid

Let $0<\delta<1$ and $\kappa\in\mathbb{N}$ with $\kappa>\frac{20}{\delta}$. For $1<q<\infty$, $0<\varepsilon,\nu<1$, let $\mathcal{E}_{\operatorname{disj}\nu}^{\delta,\kappa}(\otimes_3L^{\infty}\rightarrow L^{\frac{q}{3}};\varepsilon)$ denote the single-scale disjoint trilinear inequality defined using smooth Alpert pseudoprojections and $\nu$-disjoint triples of squares. Smooth $(\delta,\kappa)$-Alpert trilinear Fourier extension conjecture. For every $q>3$ there is $\nu>0$ depending on $q$ such that $\mathcal{E}_{\operatorname{disj}\nu}^{\delta,\kappa}(\otimes_3L^{\infty}\rightarrow L^{\frac{q}{3}};\varepsilon)$ holds. This conjecture is equivalent, by the theorem stated in the source, to the linear Fourier extension conjecture for the paraboloid in $\mathbb{R}^3$; its resolution is not supplied here.