The Kakeya maximal-function volume conjecture

Let $\mathbb{T}$ be a set of length-one $\delta$-tubes in $\mathbb{R}^n$, and let $U(\mathbb{T})$ denote their union. Assume that the tubes have $\delta$-separated directions. Kakeya maximal-function volume conjecture. For every $\epsilon>0$, there is a constant $c(n,\epsilon)$ such that, whenever $\mathbb{T}$ contains approximately $\delta^{-(n-1)}$ tubes,

$$|U(\mathbb{T})|\geq c(n,\epsilon)\delta^\epsilon.$$

This is a tube-volume formulation of the Kakeya conjecture. The source states that the corresponding Hausdorff-dimension conjecture is proved in dimensions $2$ and $3$, but the conjecture remains open in dimensions $n\geq4$.