An extremal problem on crossing vectors

For positive integers $w$ and $k$, two vectors $A$ and $B$ from ${\mathbb{Z}}^w$ are called $k$-crossing if there are two coordinates $i$ and $j$ such that $A[i]-B[i]\ge k$ and $B[j]-A[j]\ge k$. What is the maximum size of a family of pairwise $1$-crossing and pairwise non-$k$-crossing vectors in ${\mathbb{Z}}^w$? We state a conjecture that the answer is $k^{w-1}$. We prove the conjecture for $w\le 3$ and provide weaker upper bounds for $w\ge 4$. Also, for all $k$ and $w$, we construct several quite different examples of families of desired size $k^{w-1}$. This research is motivated by a natural question concerning the width of the lattice of maximum antichains of a partially ordered set.