The weighted union-set conjecture is false

We disprove a weighted generalization of the union-set conjecture [Fra95] raised in [Gow16]. We construct nonnegative weights on subsets of a finite ground set satisfying $w(\mathrm{\varnothing })=0$ and $w(A\cup B)\ge \min \{w(A),w(B)\}$, while every coordinate belongs to an arbitrarily small fraction of the total weight.