Sample hypothesis with only minimal amount of knowledge on it.

PEs seek to make profit, and are looking for places where they can either raise prices or lower costs (which will quickly correlate with worse outcomes) while not losing customers (yes, you could call them patients, but PE will view them as customers), or at least losing so few that the overall numbers result in more profit. One way of doing this is looking for barriers to competition/moats. CON is just one type of moat, and so is one factor PEs evaluate, but the presence or absence of other moats can still override the presence or absence of this one moat. One could try to work this out from data with some sort of regression, but with so many possible moats and a relatively limited number of data points, it would be easy to overfit the data.

In comparison, non-PE hospitals might have some profit motive (or keeping to budgets, not going bankrupt, ect.), but will be less driven by this mentality and thus their relationships to moats will be more complex, and so something like a CON requirement won't be as fully exploited to raise prices or lower costs.

This also fails to account for other ways that PE can seek to make money, which involves more complex parts of law and financing that I'm not well versed on (I've ready some things about real estate, but don't know enough to fairly analyze the claims).