The 1-2-3 Conjecture on neighbour sum distinguishing edge colourings

Let $G$ be a graph with no isolated edges. For an edge colouring $c:E(G)\to\{1,2,\ldots,k\}$, define the weighted degree of a vertex $v$ by

$$s_c(v):=\sum_{u\in N(v)}c(uv).$$

The colouring is neighbour sum distinguishing if $s_c(u)\neq s_c(v)$ for every adjacent pair $u,v$.

1-2-3 Conjecture. Every graph $G$ containing no isolated edges admits a neighbour sum distinguishing $3$-edge colouring.

A neighbour sum distinguishing $5$-edge colouring is known for every graph without isolated edges, but whether three colours always suffice remains open. The conjecture is equivalent to seeking a locally irregular multigraph obtained by replacing each edge with a number of parallel edges from $1$ to $3$.