The general Kakeya conjecture

Let $E$ be an arbitrary subset of $\mathbb{R}^n$, and let $D$ be the set of directions in which $E$ contains a line segment. General Kakeya conjecture. If $D$ is non-empty, then

$$\dim_H E\geq \dim_H D+1.$$

The paper proves that this conjecture is equivalent to the Kakeya conjecture. The preceding discussion notes that the conjecture would imply the expected lower bounds for Besicovitch sets, while the general Kakeya conjecture itself remains unresolved.