Four-parts conjecture for biregular bipartite labeled multigraphs

Let $L_1,R_1,L_2,R_2$ be vertex sets of sizes $n_1,k_1,n_2,k_2$, respectively, satisfying

$$n_1k_2=n_2k_1.$$

Let $(L_1,R_1,E_1,u_1,v_1)$ and $(L_2,R_2,E_2,u_2,v_2)$ be biregular bipartite labeled multigraphs with $|E_1|=|E_2|=n_1k_2$. For functions with common domain, let $[u,v]$ denote the weighted complete bipartite graph or matrix determined by the multiplicities of labels, and let $W_{L,R}$ denote the all-ones weighted complete bipartite graph on $L$ and $R$. Four-parts conjecture. There exists a bijection $\psi:E_1\to E_2$ such that

$$[u_2\circ\psi,v_1]=W_{L_2,R_1}$$

and

$$[u_1\circ\psi^{-1},v_2]=W_{L_1,R_2}.$$

The conjecture is formulated in the same spirit as the balls conjecture and the paper proves a corollary and special cases; the full statement remains open.