The generalized Kakeya conjecture

Let $E\subseteq\mathbb{R}^n$ and let $D\subseteq\mathbb{RP}^{n-1}$ be the set of directions in which $E$ contains a line segment. Write $\operatorname{hdim}$ for Hausdorff dimension. Assume that $D\neq\varnothing$.

Generalized Kakeya conjecture.

$$\operatorname{hdim} D+1\leq\operatorname{hdim} E.$$

The claim relates the dimension of the directions represented by line segments in $E$ to the dimension of $E$. In the source it is presented as equivalent to the Kakeya conjecture and is not assigned a resolution status; the status is therefore recorded as open.