The multilinear Kakeya conjecture

For $j=1,\ldots,n$, let $\mathcal{T}_j$ be a family of $\delta$-tubes in $\mathbb{R}^n$. Call the families transversal when, for each $j$, every tube in $\mathcal{T}_j$ points in a sufficiently small fixed neighborhood of the $j$th standard coordinate direction. Multilinear Kakeya conjecture. For each $n/(n-1)\leq q\leq\infty$, there is a constant $C$ such that, for all $\delta>0$ and all transversal families,

$$\left\|\prod_{j=1}^n\left(\sum_{T_j\in\mathcal{T}_j}\boldsymbol{1}_{T_j}\right)^{1/n}\right\|_{L^q(\mathbb{R}^n)}\leq C\delta^{n/q}\left(\prod_{j=1}^n\#\mathcal{T}_j\right)^{1/n}.$$

This multilinear Kakeya estimate was introduced as a route to the multilinear restriction conjecture and is known in important endpoint or special settings, but the source presents it as a conjectural principle.