Optimal list threshold for majority edge-colourings

Prove that every graph with minimum degree at least k^2 admits a 1/k-majority edge colouring from every assignment of lists of size k+1, for every integer k≥2.

Background

The paper proves a list-colouring result with minimum-degree threshold 2k^2−2k and observes that the threshold can be lowered to approximately k^2−k for certain regular graphs. This motivates a direct strengthening of the non-list conjecture to arbitrary list assignments.

The proposed statement would determine the optimal order and, up to the known obstruction at degree k^2−1, the exact threshold for 1/k-majority edge colourings from lists of size k+1.