The (2,2)-Conjecture for graph decompositions

Let $G$ be a connected graph of order at least four. A locally irregular multigraph is a multigraph in which the endpoints of every edge have different degrees. For a subgraph $H$ of $G$, let ${}^2H$ denote the multigraph obtained by replacing each edge of $H$ by two parallel edges.

(2,2)-Conjecture. The graph $G$ can be decomposed into two subgraphs $G_1$ and $G_2$ such that there exist locally irregular multigraphs $M_1$ and $M_2$ satisfying

$$G_1\subseteq M_1\subseteq {}^2G_1,$$ $$G_2\subseteq M_2\subseteq {}^2G_2.$$

Equivalently, every connected graph of order at least four should have a $(2,2)$-coloring. This formulation connects locally irregular decompositions with neighbor-sum-distinguishing edge colorings; the source presents it as an open conjecture.