Linear-diameter conjectures for list-recolouring graphs

Let $G$ be a connected graph with $n$ vertices, and let $L$ be a list-assignment of $G$. Write $\mathcal{C}(G,L)$ for the $L$-recolouring graph and $\widehat{\mathcal{C}}(G,L)$ for the corresponding relaxed recolouring graph. Linear-diameter conjectures. (a) If, in addition to the conditions in the Key Lemma, $G$ has minimum degree at least $3$, then $\mathcal{C}(G,L)$ is connected and has diameter $O(n)$. (b) If, in addition to the conditions in the Main Theorem, $G$ has minimum degree at least $3$, then $\widehat{\mathcal{C}}(G,L)$ is connected and has diameter $O(n)$. (c) If, in addition to the conditions in the Main Theorem, $G$ has no path of degree-$2$ vertices of length more than $t$ for some $t\geq0$, then $\widehat{\mathcal{C}}(G,L)$ is connected and has diameter $O((t+1)n)$. These conjectures seek linear, or controlled near-linear, diameter bounds beyond the quadratic diameter forced by long induced paths of degree-$2$ vertices. The supplied passage does not provide evidence that any of the three claims has been resolved.