Petrov's balls conjecture for biregular labeled multigraphs

Let a bipartite labeled multigraph be a tuple $(L,R,E,u,v)$, where $u:E\to L$ and $v:E\to R$. Let $A_{n,k}$ be the set of biregular bipartite labeled multigraphs with $L=[n]$, $R=[k]$, and $E=[n]\times[k]$; let $I$ be the set of involutions in the permutation group of $[n]\times[k]$. For functions $u,v$ with common domain $E$, let $[u,v]$ denote the weighted $L\times R$ matrix whose $(x,y)$ entry counts edges $e$ with $u(e)=x$ and $v(e)=y$, and let $W_{[n],[k]}$ be the all-ones matrix. Petrov's balls conjecture. For every $(u,v)\in A_{n,k}$, there exists an involution $\iota\in I$ such that

$$[u\circ\iota,v]=W_{[n],[k]}.$$

In the balls-and-colors interpretation, this asserts that the balls can be paired or left fixed so that every girl ends with exactly one ball of each color, with every ball involved in at most one swap. The conjecture implies the weak balls conjecture, but the general statement remains unproved; special cases are established in the paper.