The list strengthening of the minimum-degree conjecture

Let $G=(V,E)$ be a graph. An $\ell$-list assignment is a map $L:E\to 2^C$ with $|L(e)|=\ell$ for every edge. An $L$-colouring is an edge colouring using a colour from each edge's list, and it is a $1/k$-majority edge colouring if $d_{E_c}(v)\leq d(v)/k$ for every vertex $v$ and colour $c$.

List strengthening conjecture. For every integer $k\geq 2$, if a graph $G$ has minimum degree $\delta(G)\geq k^2$, then $G$ has a $1/k$-majority edge colouring from any lists of size $k+1$.

This directly strengthens the minimum-degree conjecture for unrestricted colours and is supported by the regular-graph result proved in the paper. The general statement remains open.