Efficient sampling conjecture for Potts models on random regular bipartite graphs

Let $G$ be a random $\Delta$-regular bipartite graph on $2n$ vertices, let $\mu_{G,q;\beta}$ be the anti-ferromagnetic $q$-state Potts Gibbs distribution, and let an FPAUS mean a fully polynomial-time almost-uniform sampler. Efficient-sampling conjecture. For $\beta>0$, with high probability there exists an FPAUS for $\mu_{G,q;\beta}$ running in time polynomial in $n$ and $\Delta$. Although single-site Glauber dynamics can mix torpidly, whether another fast Markov chain, such as polymer dynamics, yields an FPAUS remains open.