Modulated single-scale Fourier square function disjoint trilinear extension conjecture

Let $U_{1},U_{2},U_{3}\subset U$ be a $\nu$-disjoint triple of squares, meaning that each has side length comparable to $\nu$ and distinct squares are separated by at least $\nu$. For $1<q<\infty$ and $0<\varepsilon,\nu<1$, let $\mathcal{A}_{\operatorname*{disj}\nu}^{\operatorname*{square}}\left(\otimes_{3}L^{\infty}\rightarrow L^{\frac{q}{3}};\varepsilon\right)$ denote the assertion that there is a positive constant $C_{q,\varepsilon,\nu}$ such that

$$\left\Vert\mathcal{S}_{\operatorname*{Fourier}}^{s,\mathbf{u}_{1}}f_{1}\,\mathcal{S}_{\operatorname*{Fourier}}^{s,\mathbf{u}_{2}}f_{2}\,\mathcal{S}_{\operatorname*{Fourier}}^{s,\mathbf{u}_{3}}f_{3}\right\Vert_{L^{\frac{q}{3}}(\mathbb{R}^{3})}\leq C_{q,\varepsilon,\nu}2^{\varepsilon s}\prod_{k=1}^{3}\left\Vert f_{k}\right\Vert_{L^{\infty}(U)}$$

for all $s\in\mathbb{N}$ with $2^{-s}\leq\nu$, all $f_{k}\in L^{\infty}(U_{k})$, all sequences $\mathbf{u}_{k}\in\mathcal{V}$, and all $\nu$-disjoint triples. The modulated single-scale Fourier square function disjoint trilinear extension conjecture. For every $q>3$ there is $0<\nu<1$ such that $\mathcal{A}_{\operatorname*{disj}\nu}^{\operatorname*{square}}\left(\otimes_{3}L^{\infty}\rightarrow L^{\frac{q}{3}};\varepsilon\right)$ holds for all $0<\varepsilon<1$. This is the trilinear, separated-frequency counterpart of the preceding square-function conjecture and is intended to capture the multilinear structure relevant to the Kakeya equivalence.