The $n$-dimensional Minkowski arithmetic Kakeya conjecture

Fix a positive integer $n$. Let $\mathcal{f}^M_n(k)$ be the infimum of the Minkowski dimension of a set $B_k\subset\mathbb{R}^n$ containing a $k$-term arithmetic progression with every common difference in $[0,1]^n$. $n$-dimensional Minkowski arithmetic Kakeya conjecture.

$$\lim_{k\to\infty}\mathcal{f}^M_n(k)=n.$$

This is one of three higher-dimensional formulations that the source says are equivalent. It asserts that increasingly long progressions with all common differences force Minkowski dimension approaching the ambient dimension; no resolution is given.