Weakly intersecting family in a product alphabet ($S(x)=\{i:x_i>1\}$)

Here two words must both have symbols greater than one in some common coordinate, without requiring the symbols to agree. Their supports $S(x)$ therefore form an intersecting family. The fibre of a support of size $r$ contains $(k-1{)}^r$ words, relating this problem to a weighted set family problem.