Mostly just inside reports. Proof is a tricky thing in the NFL. Is there really any proof that the Rooneys feuded with Terry Bradshaw over his decision to retire (right after assuring them that he would be sticking around for at least five more years and there was no need to draft Dan Marino)? No. But that doesn't mean it didn't happen. In fact, it is pretty commonly known that it did happen.
It is a similar situation here. Hell, the Post Gazette even ran a story in 2009 stating Arians was to be fired. The reporter (can't recall who at the moment, but I remember that it was a pretty credible one), still backs the story and says that Roethlisberger made a last minute appeal on Arians' behalf and saved his job. Another example is that it has been acknowledged in interviews by Arians that he planned to "retire," after 2010, until Roethlisberger "talked him out of it." Inside sources however, told local sports reporters like Guy Junker, Bob Pompeani and Mark Madden that Arians was going to be fired for badmouthing Tomlin to players until Roethlisberger intervened and talked to Rooney and Colbert.
Obviously, no one from the front office goes to the press and says "Yep, we were going to fire that son of a ***** but Roethlisberger likes him so we didn't." But again, that doesn't mean that it isn't happening. The reporters who have discussed it (Junker, Pomp, etc) are all pretty reliable guys. They wouldn't put their reputations on the line unless they have sources that they damn well trust.
That all comes down to Arians, well...and Tomlin. He is the only coordinator in the NFL who throws more than he runs with the lead in the fourth quarter (I don't count the Pats as Belichick loves to run up the score

). Arians' idea of increasing the run attack means trying it a few times...but the second that even one run does not work, he will completely abandon the run game. I have never seen a coordinator who lacks such basic football knowledge like, "throw less in 40 mph winds," or "run the ball with the lead late in the game." Of course, this is Tomlin's responsibility too. He can overrule Arians any time that he wants....but he doesn't.