[gautampk]

I think the naive expectation would be that a hot state would be highly entangled and non-separable. The non-separability is what leads to decoherence when you do a partial trace. You can arbitrarily split any multi-part Hilbert space into a "system" and an "environment" and then trace out the environment. If they are highy entangled with one another then the resulting system-without-environment will be decohered.

[gradschoolfail]

Alternatively formulated, this suggests that an entangled state, if weakly coupled to a bath at high temperature and allowed to thermalize, will become fully disentangled at a finite time. (p1)

hinting that something different is happening here (from the C&L model mentioned below.)

Indeed the CS guys are probably looking for a way to redefine entanglement (entropy, as you summarized above), to essentially “work” in the same way as what they have in their preprint, but also cover the prethermalized cases..