The claim was that school teacher salaries were low "Because of targeted propaganda and attacks by an organized political group that holds, as an ideology, that public institutions should not exist and everything should be done by the private sector.
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Since private schools are not subject to these attacks, their salaries would not be lowered by these attacks, making them relatively higher.
Yeah, the horrible pay in private school is proof enough that the "free" market is the absolute wrong way to go if you want better teachers.