I kind of think you've got it backwards with regards to the tape. According to her, he asked her not to fly a flag,and according to her she deferred. But he says the exact opposite, and claims that she flew it in spite of him asking her not to.
She also said that she only had to outlast (her opposition) for 5 yrs, and then she was free. The implication was that Alito will retire (presumably during a Trump Presidency). So it sounds like they're banking on a Trump win...
And there's nothing wrong with that, except for the fact that it suggests Alito has a vested interest in Trump winning. Which to a lot of folks means Alito should probably recuse himself from SCOTUS cases that involve Trump and consequently could affect the election...
There are other tapes of Alito recorded during the SC Historical Society dinner where rich donors get to hobknob with right wing SCOTUS justices. He tells this second woman that ProPublica is to blame for their scrutiny on him and Thomas and that it's partisan.
But according to the reporter from Pro Publica featured in this video they just stumbled on to these facts by accident. They were doing a report concerning what SCOTUS Justices do when they aren't in session and have all that free time, and they came upon these travel logs which showed an unknown Justice taking frequent expensive trips. That's when they started to dig, and realized thru the process of elimination that it was Thomas...
It was kind of hilarious to hear Alito whine about Pro Publica's "donors" considering Crow has virtually subsidized Thomas over the years...
Here is a more indepth look at the bar graph showing the benefits to SCOTUS Justices that is shown in the video...