Consider a free quantum field theory defined upon a static Lorentzian spacetime possessing a bifurcate Killing horizon, such as Schwarzschild spacetime.
These assumptions are sufficient to define a unique quantum state, the "Hawking-Hartle vacuum", which is stationary under Schrodinger evolution by the Killing observers, and which does not diverge should that Schrodinger evolution be continued through either event horizon. When decomposed into incoming and outgoing modes, one finds emission from and absorption by the black hole are exactly balanced.
This would seem to suggest that this state should exert no gravitational back-reaction (e.g. that the black hole should not evaporate). However, computations of the renormalized stress-energy tensor find a nonzero result.
So in the Hawking-Hartle vacuum, does the black hole evaporate or not? Rephrasing: in semiclassical gravity (e.g. the use of the expectation value of the renormalized stress-tensor as a source term to the Einstein equations), does the Schwarzschild-plus-Hawking-Hartle vacuum solution continue to be static?
P.S. I am aware that the evaporation of black holes is due to the Unruh, not the Hawking-Hartle, vacuum. This question is about the latter state specifically.