Complexity of determining the irregular chromatic index of a graph

A graph G is locally irregular if adjacent vertices of G have different degrees. A k-edge colouring φ of G is locally irregular if each of the k colours of φ induces a locally irregular subgraph of G. The irregular chromatic index χirr(G) of G is the least number of colours used by a locally irregular edge colouring ofG (if any). We show that determining whether χirr(G) = 2 is NP-complete, even when G is assumed to be a planar graph with maximum degree at most 6.