I did work out the air situation many years ago when researching the Thresher incident.
Thresher had a two part problem, her test depth at 1300ft had a profound effect on her blowing her ballast during her last emergency and here is why.
Thresher's ballast systems were from an old design taken from a diesel submarine which do not dive as deep (most SSK's hand around the 1000ft mk but the earlier stuff more 500-750ft) therefore her pipe work was of smaller diameter.
As explained above not enough air could get through under enough pressure to blow the tanks one of the lessons learnt in the thresher was the pipe work had to be increased from 2in diameter to 5 inch diameter (im on memory here)
The second part of the problem was the end of the pipe was a cage to filter out FOD being so deep meant that the temperature in the water was cold (being north Atlantic very cold) and moisture from the compressors iced over in the cages which created a blockage.
So how did thresher sink?
Simple naval reasoning: The vessel sunk due to lack of buoyancy
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