The strong Kakeya maximal operator conjecture in three dimensions

Let $0<\varepsilon<1$, and let $\mathcal{K}^{\ast}(\otimes_1L^{\infty}\rightarrow L^{\frac{3}{2}};\varepsilon)$ be the statement that for every $0<\delta<1$ and every family $\mathbb{T}$ of $\delta$-separated $\delta$-tubes in $\mathbb{R}^3$, one has the corresponding $L^{3/2}$ bound for their incidence sum. Strong Kakeya maximal operator conjecture. The statement $\mathcal{K}^{\ast}(\otimes_1L^{\infty}\rightarrow L^{\frac{3}{2}};\varepsilon)$ holds for all $0<\varepsilon<1$. The claim is the strong Kakeya estimate in three dimensions; no resolution is given in the supplied text.