Recolouring threshold for triangle-free graphs

Prove that every triangle-free graph G is k-recolourable for every integer k satisfying k >= ceil((4/9)·2 + (5/9)(Delta(G) + 1)), where k-recolourability means that all proper k-colourings of G are Kempe equivalent.

Background

The paper constructs triangle-free graphs with frozen non-unique colourings at the four-ninths threshold and proves that frozen colourings below that threshold must be unique up to permutation of colours. This establishes the optimal threshold for the principal known obstruction in the triangle-free setting. The stated conjecture asks whether the absence of such frozen obstructions suffices to guarantee full Kempe recolourability at the corresponding bound.