Before revision:  We prove that, starting from a warm start, it mixes in, where f is L-smooth and m-strongly-convex, within accuracy \varepsilon after } \widetilde O(\kappa d^2 \ell^2 \log (1 / \varepsilon)) steps for a well-rounded polytope, ignoring logarithmic factors where \kappa is the condition number of the negative log-density, d is the dimension, \ell is an upper bound on the number of reflections, and \varepsilon is the accuracy parameter.
After revision:  We prove that, starting from a warm start, it mixes in, where f is L-smooth and m-strongly-convex, within accuracy \varepsilon after } \widetilde O(\kappa d^2 \ell^2 \log (1 / \varepsilon)) steps for a well-rounded convex body where \kappa = L / m is the condition number of the negative log-density, d is the dimension, \ell is an upper bound on the number of reflections, and \varepsilon is the accuracy parameter.