The $(n,k)$-set dimension conjecture

Let $1\leq k<n$, and let $E\subset\mathbb{R}^n$ be an $(n,k)$ set, meaning that for every $k$-dimensional subspace $\pi\subset\mathbb{R}^n$, there is a $k$-dimensional unit cube parallel to $\pi$ contained in $E$. The dimension may mean upper Minkowski dimension, lower Minkowski dimension, or Hausdorff dimension.

$(n,k)$-set dimension conjecture. If $E\subset\mathbb{R}^n$ is an $(n,k)$ set, then

$$\dim(E)=n.$$

This generalizes the Kakeya dimension conjecture, since an $(n,1)$ set is a Kakeya set. The paper presents the conjecture without giving a resolution status or known cases beyond this relation.