Considering the phrasing of the 1st HK theorem given in the previous topic:

“There is a one-to-one correspondence between the ground state density of a many electron system and the external potential”

Or V_ext <-> rho_gs

If the electron density is at the ground state, it seems likely that the system is at the ground case; however, this does not seem to be guaranteed by the 1st HK theorem. What situation could cause the electron density to be at the ground state, while the system overall is not? Thinking back of the discussion of when the B-O approximation fails, there may be a condition where the nuclei are energized in such a way that the electrons cannot immediately follow, as in the case of using a femto second laser pulse. So, If a system at ground state could be hit by such a laser pulse, perhaps the electron density would momentarily remain at the ground state while there is energy in the nuclei that is above the ground state. The first HK theorem does not seem to preclude such an (unlikely) scenario.