[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..