Omega-Kakeya conjecture

Let $\Omega\subset\mathbb{S}^{n-1}$ be a Borel set. A set $X\subset\mathbb{R}^n$ is an $\Omega$-Kakeya set if, for every $e\in\Omega$, it contains a unit segment oriented along $e$. Write $d_X$ and $d_\Omega$ for the Hausdorff dimensions of $X$ and $\Omega$, respectively.

$\Omega$-Kakeya conjecture. For every Borel set $\Omega\subset\mathbb{S}^{n-1}$, if $X$ is an $\Omega$-Kakeya set, then

$$d_X\geq d_\Omega+1.$$

This is the natural direction-restricted generalization of the Kakeya conjecture. The paper notes that Keleti and Mathé proved its equivalence with the Kakeya conjecture; the conjecture is therefore unresolved in the general higher-dimensional setting.