Peres–Tetali curvature-to-MLSI conjecture

Suppose $(\Omega,p)$ corresponds to lazy random walk on a finite graph, and let $d$ be the graph distance. If $(\Omega,p,d)$ has coarse Ricci curvature $\kappa>0$, let $\rho_0$ denote its modified log-Sobolev constant.

Peres–Tetali conjecture. The modified log-Sobolev constant satisfies

$$\rho_0\geq C\kappa,$$

where $C>0$ is a universal constant.

This conjecture links positive coarse Ricci curvature to exponential convergence to equilibrium in relative entropy. The source gives no evidence of a resolution, so the conjecture remains open.