Hawking radiation is a very small temperature, much less than the heat from the CMBR background of about 3 degK. Pressure is linearly proportional to temperature so the negative pressure measured inside the black hole causing it to shink slightly probably corresponds to a positive pressure outside its event horizon.
Only when the rest of the Universe reaches a temperature less than the black hole will it be able to radiate the Hawking radiation allowing the black hole to completely evaporate. This should take about a trillion years so nobody should be in a rush figure out the details.
It’s never too early to start .................