Positive-parameter recolourability for all graphs

Determine whether there exists a positive constant η such that every graph G is k-recolourable whenever k ≥ ⌈ηω(G) + (1−η)(Δ(G)+1)⌉.

Background

The paper studies Kempe-equivalence of proper colourings and recalls a question posed by Bonamy, Kaiser, and Legrand-Duchesne. The question asks whether a Reed-type lower bound on the number of colours guarantees recolourability for all graphs. The paper proves that frozen-colouring obstructions disappear below the threshold η = 1/3, but it does not establish full recolourability, so the question remains unresolved.