Before revision:  This implies that the relaxation time (i. e., the inverse of the spectral gap) is O(1) for all boundary conditions in the uniqueness region or when \beta<\beta_1 where \beta_1 exceeds the uniqueness threshold for the Ising model  and for the q-state Potts model when q is small with respect to d.
After revision:  This implies that the relaxation time (i. e., the inverse of the spectral gap) is O(1) for all boundary conditions throughout the tree uniqueness region; in fact, our bounds hold beyond the uniqueness threshold for the Ising model and for the q-state Potts model when q is small with respect to d.