Jerrum's Glauber-dynamics conjecture for graph colorings

Let $G$ be a graph of maximum degree $\Delta$, and consider the Glauber dynamics for sampling proper $q$-colorings of $G$. Jerrum's conjecture. The Glauber dynamics for sampling $q$-colorings from $G$ mixes in polynomial time whenever $q \geq \Delta+2$. Efficient Glauber-dynamics sampling at this threshold would substantially improve the best general bounds for approximate sampling of proper colorings; the paper reports that the conjecture remains open, with known results requiring more colors.