Felsner–Krawczyk–Micek crossing-vector extremal conjecture

For positive integers $k$ and $w$, let $f(k,w)$ be the maximum size of a subset of $\mathbb{Z}^w$ in which every two vectors are $1$-crossing but no two vectors are $k$-crossing. Felsner–Krawczyk–Micek conjecture. For all $k,w\ge 1$,

$$f(k,w)=k^{w-1}.$$

This is the main extremal problem of the paper. The conjecture is proved for $w\le 3$, while weaker upper bounds and constructions of families of the conjectured size are given for larger dimensions; the general case remains open.