I have no idea how reliable this source is, but it looks plausible - from the "American Investment Council", which appears to be some kind of private equity trade association ( https://www.investmentcouncil.org )

https://www.psprs.com/uploads/sites/1/AIC_PublicPensionRepor...

Some interesting details:

- "Nearly 50 percent of the private equity investment dollars that make their way into American businesses come from public pension funds", which substantiates OP's thesis.

- "U.S. public pension funds invest 9% of their portfolios in private equity, on a dollar-weighted basis." 46% is in public equity, so obviously the lion's share is in still in public markets.

This isn't surprising. Public companies tend to be lower risk (and therefore offer lower returns) than PE investments and pension funds want a mix of both. They want the juicy returns of PE deals, but a portfolio invested completely or mostly in PE would be unacceptably risky. Most pension fund mandates will set % limits on how much can be invested in different asset classes, with lower limits for riskier asset classes.