The minimum-degree conjecture for majority edge colourings

Let $G=(V,E)$ be a graph. A $1/k$-majority edge colouring with colour set $C$ is an edge colouring $\omega:E\to C$ such that, for every vertex $v$ and colour $c\in C$, the number $d_{E_c}(v)$ of incident edges coloured $c$ satisfies $d_{E_c}(v)\leq d(v)/k$.

Majority edge-colouring conjecture. For every integer $k\geq 2$, if a graph $G$ has minimum degree $\delta\geq k^2$, then $G$ is $1/k$-majority edge $(k+1)$-colourable.

The bound is best possible up to the stated threshold because, for every $k\geq 2$, there is a graph of minimum degree $k^2-1$ without such a colouring. The conjecture is known for $k=2,3,4$ and for bipartite graphs, but remains open in general.