Local Irregularity Conjecture for 2-multigraphs

Let $G$ be a simple graph, and let $^2G$ denote the 2-multigraph obtained by doubling every edge of $G$. Let $\operatorname{lir}(H)$ be the smallest number of colors in a locally irregular edge coloring of a multigraph $H$.

Local Irregularity Conjecture for 2-multigraphs. For every connected graph $G$ which is not isomorphic to $K_2$,

$$\operatorname{lir}(^2G)\leq 2.$$

The conjecture was proposed by Grzelec and Woźniak. The paper proves it for regular 2-multigraphs; its status in the generality stated here is not resolved in the supplied text.