Torpid mixing conjecture for q-coloring Glauber dynamics on high-dimensional tori

Let ${\mathbb Z}^d_n$ be the even discrete torus with vertex set $[n]^d$, where $n$ is even, and nearest-neighbor edges taken modulo $n$. Let ${\mathcal C}_q={\mathcal C}_q({\mathbb Z}^d_n)$ be the set of proper $q$-colorings, and let ${\mathcal M}_q$ denote the Glauber dynamics chain on ${\mathcal C}_q$. Write $\tau_{{\mathcal M}_q}$ for its mixing time. Torpid-mixing conjecture. For all $q>3$, all even $n\geq 2$ and all sufficiently large $d=d(q)$, the mixing time of the Glauber dynamics chain ${\mathcal M}_q$ on ${\mathcal C}_q$ is essentially exponential in $n^{d-1}$. The paper proves a rigorous exponential lower bound for $q=3$ when $d$ is sufficiently large, while the corresponding assertion for $q>3$ is left open.