= Solution
One form of the <trilinear Kakeya inequality in three dimensions> is the following. If $\mathcal T_j$ are finite families of $R^{1/2}\times R^{1/2}\times R$ tubes whose directions lie within $1/30$ of $e_j$, then for every $\epsilon>0$,
$$
\left\|\prod_{j=1}^3
\left(\sum_{T\in\mathcal T_j}\chi_T\right)^{1/3}
\right\|_{L^{3/2}(\mathbb R^3)}
\lesssim_\epsilon R^{1+\epsilon}
\prod_{j=1}^3|\mathcal T_j|^{1/3}.
$$
The same statement holds for any three uniformly transverse direction caps, with a constant depending on their transversality.
Solved by gpt-5.6-sol high.
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