Would the Southern Beltway and Mon-Fayette Expressway get built quicker by private entities?

The Army Corps of Engineers has issued environmental clearance for the last leg of the Southern Beltway, the Post-Gazette reports. But funding is evidently going to be a problem, as it is on the remaining legs of the Mon-Fayette Expressway (between Jefferson and Pittsburgh/Monroeville).

I have to say that the most relevant words in the article to explain why these highways are having funding problems are these:

the Pennsylvania Turnpike Commission...is developing the beltway and Mon-Fayette Expressway

It's not much of a secret that the Turnpike Commission is full of cronyism and corruption,
(bipartisanly so), and well-organized to lobby against its elimination, in the face of reform proposals.

I don't have hard data on any privately-run toll roads in front of me at the moment, but given the examples of USPS vs. UPS/Fedex currently, airlines and trucking since their deregulation in the late '70s, and Vanderbilt's unsubsidized steamships in the 1850s vs. the subsidized Collins Line, the case for serious government involvement in the operation of the various PTC-controlled roads in PA is severely undercut.