List-colouring characterization of layers

Determine for which values of k every collection of k-element colour lists on the edges of K_{n,n}, in which each colour appears in exactly k incident edge lists at every vertex, admits a proper list-colouring of K_{n,n}.

Background

The paper reformulates the existence of a layer in an (n²,k)-array as a list-colouring problem on the complete bipartite graph K_{n,n}. Each edge receives a list of exactly k colours, and every colour occurs in exactly k lists incident with each vertex. A proper colouring from these lists corresponds to a Latin square and hence to a layer in the array. The authors ask for the complete range of k for which such a colouring is guaranteed.

References

Question 1. Let Kn,n be such that for each edge e ∈ E(Kn,n) there exists a list Le of exactly k colours in {1, . . . , n} such that for every i ∈ {1, . . . , n} and every vertex v ∈ Kn,n there exist exactly k edges e incident to v such that i ∈ Le. For which values of k does every such collection of lists admit a proper colouring of Kn,n from these lists?

— Extendibility of Latin Hypercuboids  (2502.08868 - Bowtell et al., 13 Feb 2025) in Question 1, Section 3, p. 7