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.