The Kakeya conjecture in maximal-function form

Fix $n\geq 2$ and $0<\delta\ll 1$. Let $\Omega$ be a maximal $\delta$-separated subset of directions on $S^{n-1}$, and for each $\omega\in\Omega$ let $T_\omega$ be a $\delta\times 1$ tube in direction $\omega$ inside the ball. Given $0<\lambda\leq 1$, let $\widetilde T_\omega\subseteq T_\omega$ satisfy $|\widetilde T_\omega|=\lambda|T_\omega|$. Write $X\lessapprox Y$ for $X\leq C_\varepsilon\delta^{-\varepsilon}Y$ for every $\varepsilon>0$. Let $K_X(p,n)$ denote the assertion that

$$\left|\bigcup_{\omega\in\Omega}\widetilde T_\omega\right|\gtrapprox\lambda^p\delta^{n-p}$$

for all such choices. Kakeya conjecture, maximal version. We have $K_X(p,n)$. The estimate $K_X(1,n)$ is trivial, and the conjectural difficulty is to make $p$ large enough to prevent excessive overlap of the selected tube subsets. The maximal version is stronger than the Hausdorff and Minkowski versions because it also addresses small $\lambda$; in three and higher dimensions the conjectures remain open.