May i dare to change one thing in swampthings post which is the correlation between TAS and Mach. I think what you meant to say was IAS and Mach, since that is what the coffin corner charts are based on. The correlation between TAS and Mach is merely dependent on temperature and yes in an ideal atmosphere the temperature would go down between FL370 and FL410 so on constant Mach the TAS would decrease. But if we are talking northern hemisphere the tropopause would typically set in at an altitude equivalent of the FL's we are talking about now.
The IAS would decrease anyway and that's where Mmo starts going towers V(IAS)stall