Sequential orderability of all nonexceptional connected graphs

Prove that every proper edge colouring of every connected graph other than $K_2$ or an even cycle is sequentially orderable; equivalently, determine a global total order of the edges that induces distinct incident-edge weight sequences at the endpoints of every edge.

Background

The paper fixes a proper edge colouring of a connected graph and asks whether the edges can be given a single global total order such that the sequences of incident edge weights distinguish every pair of adjacent vertices. The authors prove this whenever two adjacent vertices have different palettes, which covers connected non-regular graphs and connected graphs of class two. They also establish the result for connected regular graphs of degree at least six using a probabilistic argument.

The remaining cases are proper edge-colourings of 2-connected $d$-regular graphs for $d\in\{3,4,5\}$. The paper identifies $K_2$ and properly 2-edge-coloured even cycles as genuine exceptions and conjectures that all other connected graphs admit the required edge order.