Given two positive integers $n, m$, let $A$ be a multiset of $n$ integers taken from { $ 1,2,\cdots, m$ }, and $B$ be a multiset of $m$ integers taken from { $1,2,\cdots,n$ }.

Is it always possible to choose two nonempty subsets from $A$ and $B$ respectively, such that their sums are equal?

I cannot either prove or disprove it. Trying some examples made me conjecture that it is true.