Flandrin et al.'s neighbour sum distinguishing index conjecture

Let $G$ be a connected graph of order at least three, different from the cycle $C_5$. Write $\chi'_{\sum}(G)$ for its neighbour sum distinguishing index and $\Delta(G)$ for its maximum degree. Flandrin et al.'s conjecture.

$$\chi'_{\sum}(G) \leq \Delta(G)+2.$$

This conjecture concerns proper edge colourings in which adjacent vertices have distinct sums of incident edge colours. It was verified for several classical graph families, including paths, cycles, complete graphs, complete bipartite graphs and trees; its general status is not resolved in the supplied source.