Three-subgraph regular-irregular decomposition conjecture

A subgraph is called regular or locally irregular according as it is regular or locally irregular. The regular-irregular number $\operatorname{reg\text{-}irr}(G)$ is the minimum number of subgraphs into which the edges of a graph $G$ can be decomposed, with every subgraph regular or locally irregular. Three-subgraph regular-irregular conjecture. Every graph can be decomposed into $3$ subgraphs, such that each subgraph is locally irregular or regular.

This conjecture weakens the conjecture that every non-exception graph has irregular chromatic index at most $3$ and strengthens the conjecture that every graph has regular-irregular chromatic index at most $2$. The supplied text does not report a resolution.