uncommon questions - tachyons and negative temperatures

Negative absolute temperatures occur in systems that are bounded in energy from above, implying that systems with free particles can, due to their dispersion relation with zero as the lowest kinetic energy, only have positive absolute temperatures. Could one set up a system consisting of a thermal ensemble of free tachyons that has negative absolute temperature?

bonus question: can one set up a gravitationally bound system of thermal tachyons at negative absolute temperature?