Given $k$ girls, they are given $kn$ balls so that each girl has $n$ balls. Balls are coloured with $n$ colours so that there are $k$ balls of each colour. Two girls may exchange the balls (1 ball for 1 ball, so each girl still has $n$ balls), but no ball may participate in more than one exchange. They want to achieve the situation when each girl has balls of all $n$ colours. Is it always possible?

On other language. Given is a bipartite multigraph $G=(V_1,V_2,E)$, $|V_1|=k$, $|V_2|=n$, each vertex in $V_1$ has degree $n$ and each vertex in $V_2$ has degree $k$. We may replace two edges $ab,cd$ ($a,c\in V_1, b,d \in V_2$) to $ad,cb$, but new edges can not be used in exchanges anymore. Is it possible to get a usual $K_{k,n}$ without multiplicities?

If yes, this implies the positive answer to this question, which I find quite interesting itself.

I think I may prove it when $\min(n,k)\leqslant 3$, but already for $3$ there are many cases to consider.