The possibility of symmetry breaking of the fluid (argon) density distribution across a long closed slit with identical walls composed of solid carbon dioxide was noted in previous papers by the authors. The main conclusion was that there is a range of average densities in which symmetry breaking occurs and that outside that range the fluid density profile is symmetrical. A critical temperature T-sb was also identified below which symmetry breaking can occur. In this paper, symmetry breaking is examined for walls made of other materials and it is shown that it occurs only when the energy parameter epsilon(fw) of the fluid-wall interaction in the Lennard-Jones potential satisfies the inequalities epsilon(fw1) <= epsilon(fw) <= epsilon(fw2), where epsilon(fw1) and epsilon(fw2) are temperature-dependent critical values of cf,. The value of cfwl increases and that of epsilon(fw2) decreases with increasing temperature. The comparison of the theory with Monte Carlo simulations confirms the existence of symmetry breaking across the slit. The possibility of symmetry breaking along the slit is also noted.