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Yeah, it's a Maxwell’s demon with an extra phase change involved. Nobody's figured out that "passive separation" step in detail, and the second law basically states it's not just our lack of imagination. You seem to know that, though.
Think about it, you have your vapour in the top of the vessel at equilibrium with the liquid, and you close a valve to cordon it off. Then what? You pull it out and leave a vacuum in it's place? That change in pressure is going to involve quite a bit of work. There's also conduction that's working against you, so I'm not even certain the vapour is hotter to start with.
It's worth mentioning that unlike the other laws of physics, the laws of thermodynamics are emergent, and nobody's rigorously proven they actually work as described without some kind of handwave. The thing is, nobody doubts thermodynamics - the empirical evidence is mountainous and the almost-proofs are convincing. The missing part probably has to do with P and NP and pseudorandomness, which is one of the millennium problems.
Sir Arthur Stanley Eddington, The Nature of the Physical World (1927)