Kakeya conjecture for Hausdorff, packing and Minkowski dimension

A Besicitch set is a subset $B\subset\mathbb{R}^n$ containing a unit line segment in every direction. For such a set, let ${\mathrm{dim}}\xspace_H B$, ${\mathrm{dim}}\xspace_P B$, and ${\mathrm{dim}}\xspace_M B$ denote its Hausdorff, packing, and Minkowski dimensions, respectively. Kakeya conjecture. If $B$ is a Besicovitch set, then

$${\mathrm{dim}}\xspace_H B=n,\qquad {\mathrm{dim}}\xspace_P B=n,\qquad {\mathrm{dim}}\xspace_M B=n.$$

The three assertions are ordered by implication through ${\mathrm{dim}}\xspace_H B\leq{\mathrm{dim}}\xspace_P B\leq{\mathrm{dim}}\xspace_M B$; none of the converse implications is known. The conjecture is a central dimension problem concerning sets containing line segments in every direction.