Recolouring conjecture for triangle-free graphs

Let $G$ be a triangle-free graph, let $\Delta(G)$ be its maximum degree, and let $k$ denote the number of colours in a proper colouring. A colouring is Kempe-recolourable when it can be transformed by Kempe changes, interchanging two colours on a connected bichromatic component.

Triangle-free recolouring conjecture. Any triangle-free graph $G$ is $k$-recolourable for all

$$k \geqslant \left\lceil \frac{4}{9}\cdot 2 + \frac{5}{9}(\Delta(G) +1) \right\rceil.$$

The threshold is motivated by the paper's optimal analysis of frozen-colouring obstructions in triangle-free graphs, and the conjecture is tight in view of constructions above the threshold. The recolouring statement itself remains open.