The sticky Kakeya conjecture

Let $\mathcal{L}$ be the set of affine lines in $\mathbb{R}^n$, with packing dimension used for subsets of this line space. A compact set $K\subset\mathbb{R}^n$ is a sticky Kakeya set if there is a set of lines $L\subset\mathcal{L}$ with packing dimension $n-1$, containing at least one line in each direction, such that $\ell\cap K$ contains a unit interval for every $\ell\in L$. The sticky Kakeya conjecture. Every sticky Kakeya set in $\mathbb{R}^n$ has Hausdorff and Minkowski dimension $n$. This is a special case of the Kakeya conjecture, motivated by the multiscale self-similarity of sticky configurations. The paper proves it in three dimensions, while it remains open in three and higher dimensions as a general statement.