Per this story, it mentioned several times that the meet was on her plane - there appears to have been no ‘neutral’ 3rd plane. The only time I’ve heard about a 3rd plane was a mention from Rush - frankly, the way some of the earlier stories were written (poorly) it would be easy to get confused.
If there had been a 3rd/neutral plane, or even Lynch’s plane sitting on the tarmac, cabin air & electric power can be run by the APU (auxilliary power unit) without the main engines being started so it would not have been hot inside.
Roger.