The multigraph formulation of the (2,2) Conjecture

Let $G$ be a connected graph of order $n\geq 4$. Let $\mathcal{M}^{[2]}(H)$ denote the family of multigraphs obtained from a graph $H$ by edge multiplication with edge multiplicities at most $2$, and call a multigraph locally irregular when adjacent vertices have distinct degrees. (2,2) Conjecture. The graph $G$ can be decomposed into two subgraphs $G_r$ and $G_b$ such that there exist locally irregular multigraphs

$$\hat G_r\in\mathcal{M}^{[2]}(G_r),\qquad \hat G_b\in\mathcal{M}^{[2]}(G_b).$$

This is the multigraph formulation of the preceding $(2,2)$-coloring conjecture. Its general status is open.