Fare integration has nothing to do with the engineering and capitalization required with thru-running, who some trivilize by drawing pretty and orderly Vignelli style subway maps for commuter rail, assuming Amtrak is converting to 60 cycle (they're not), and waving a magic wand telling everyone to buy universal rolling stock.
NJT and PATH could have gone along with MTA E-tix and OMNY, but they are each going their own way. So much for that.
Since LIRR's 63rd Street tunnels to Grand Central Madison can only clear flat top 3rd rail MU's that are 12'10" in height, will run 50% of their electric service there, and their trains circulate among multiple branches and city terminals every day, to tell them to buy tri-voltage MU's is a fool's errand.
To save NJT the cost of Penn Station South, any expectation for LIRR to disrupt their operations and equipment rosters, "widen the platforms" (Translation - rip out half the NYPS tracks that NJT does not own) when LIRR has nothing to do with New Jersey, and already thru-run their trains to the West Side Yard, seldom occupying NYPS for more than 5 or 10 minutes - is all downright funny.