Recolouring conjecture for odd-hole-free graphs

Let $G$ be an odd-hole-free graph of maximum degree $\Delta$ and clique number $\omega$. A proper colouring is Kempe-recolourable when it can be transformed through Kempe changes, which interchange two colours on a connected component of the corresponding bichromatic subgraph.

Odd-hole-free recolouring conjecture. All odd-hole-free graphs of maximum degree $\Delta$ and clique number $\omega$ are $k$-recolourable for

$$k \geqslant \left\lceil \frac{\omega + \Delta + 1}{2}\right\rceil.$$

This is motivated by known Kempe-recolouring results for odd-hole-free graphs, which rule out the main known frozen-colouring obstruction above the local Reed threshold. The conjectured full recolourability at this sharper bound remains open.