Felsner–Krawczyk–Micek generalized crossing-vector conjecture

For $w\ge 1$ and integers $1\le k_1\le\cdots\le k_w$, say that two vectors $A,B\in\mathbb{Z}^w$ are $(k_1,\ldots,k_w)$-crossing if there are coordinates $i,j$ such that $A[i]-B[i]\ge k_i$ and $B[j]-A[j]\ge k_j$. Let $f(k_1,\ldots,k_w;w)$ be the maximum size of a subset of $\mathbb{Z}^w$ in which every two vectors are $1$-crossing but no two vectors are $(k_1,\ldots,k_w)$-crossing. Felsner–Krawczyk–Micek generalized conjecture. For $w\ge 1$ and $1\le k_1\le\cdots\le k_w$,

$$f(k_1,\ldots,k_w;w)=k_2\cdots k_w.$$

The paper notes that this apparently more general conjecture is equivalent to the main conjecture, since $f(k,w)=f(k,\ldots,k;w)$. The general case remains open in the supplied text.