Once in space, each was found to have a unique response to variations in direct solar heating. (nap.edu)
The condition that portions of the upper atmosphere are significantly warmer than otherwise via direct solar heating should (so far as I know) enhance the GHG - induced cooling at those levels. (realclimate.org)
But without direct solar heating of the stratosphere (and in the approximation of no non-radiative energy fluxes throught the stratosphere), while the temperature should decline through the stratosphere, it will not go to zero. (realclimate.org)