The Kakeya projection conjecture

Let $K\subset\mathbb{R}^n$ be a Kakeya set, and let $G(n,k)$ denote the Grassmannian of $k$-dimensional subspaces of $\mathbb{R}^n$. For $\gamma\in G(n,k)$, write $\pi_\gamma$ for orthogonal projection onto $\gamma$, and let $\dim_{\mathrm{H}}$ denote Hausdorff dimension. Projection conjecture. For every integer $0<k<n$, there is a number $c$ such that for all $\gamma\in G(n,k)$,

$$\dim_{\mathrm{H}}(\pi_\gamma K)=c.$$

The conjecture proposes direction-independent Hausdorff dimension for all orthogonal projections of every Kakeya set. The paper states that this projection property is equivalent to the Kakeya conjecture, so its resolution is tied to the open Kakeya dimension problem.