Polynomial mixing-time conjecture for Glauber dynamics on graph colourings

Let $G$ be a graph with maximum degree $4\Delta4$, and let $k$ be an integer satisfying

$$k \geq \Delta+2.$$

Consider the Glauber dynamics on the set of $k$-colourings of $G$, in which a uniformly random vertex is recoloured with a uniformly random available colour.

Polynomial mixing-time conjecture. For any graph $G$, the Glauber dynamics on the $k$-colourings of $G$ for $k\geq\Delta+2$ has polynomial mixing time.

This conjecture asserts rapid mixing at the standard threshold of two more colours than the maximum degree. Polynomial mixing is important because it would make Glauber dynamics an efficient sampler for uniformly random proper colourings; the source presents this as a well-known conjecture and does not state a resolution.