Regular Cereceda's Conjecture

Let $G$ be a $d$-regular graph, let $\mu(G)$ denote its matching number, let $n(G)$ denote its number of vertices, and let $\mathcal{C}_k(G)$ be the reconfiguration graph of proper $k$-colourings of $G$.

Regular Cereceda's Conjecture. If $k=d+2$, then

$$\operatorname{diam} \mathcal{C}_k(G)=n(G)+\mu(G).$$

This would combine the conjectured upper bound for list-colouring reconfiguration with the corresponding lower-bound question to determine the precise diameter at the threshold for regular graphs. It remains open.