Dual form of the Kakeya maximal operator conjecture in three dimensions

Let $0<\delta<1$. A $\delta$-tube is a tube of length $1$ and cross-sectional diameter $\delta$, and a family $\mathbb{T}$ of $\delta$-tubes is $\delta$-separated when distinct tube orientations differ by at least $\delta$. For $0<\varepsilon<1$, let $\mathcal{K}^{\ast}\left(\otimes _{1}L^{\infty}\rightarrow L^{\frac{3}{2}};\varepsilon\right)$ denote the assertion that there is a positive constant $C_{\varepsilon}$ such that

$$\left\Vert \sum_{T\in\mathbb{T}}\mathbf{1}_{T}\right\Vert _{L^{\frac{3}{2}}\left(\mathbb{R}^{3}\right)}\leq C_{\varepsilon}\delta^{-\varepsilon}$$

for every family $\mathbb{T}$ of $\delta$-separated $\delta$-tubes in $\mathbb{R}^{3}$ and every $0<\delta<1$. The dual Kakeya maximal operator conjecture. The statement $\mathcal{K}^{\ast}\left(\otimes _{1}L^{\infty}\rightarrow L^{\frac{3}{2}};\varepsilon\right)$ holds for all $0<\varepsilon<1$. This is stronger than the Kakeya set conjecture, which has recently been proved in $\mathbb{R}^{3}$, but the corresponding maximal-operator estimate remains unresolved.