I don't think it's staged to be fair, the two of them are just argumentative by nature. You could see Carragher starting to lose his cool on the Everton segment. The two of them care too much about being seen as knowledgeable to come out with click bait material.
I also think Carragher is a good pundit and I'm not just saying that because he said we're playing the best football in Europe along with Napoli. May not always be right with his conclusions but he at least tries to look beyond the superficial nonsense like most pundits do.
Neville has always had this sneering attitude towards us, but it's gone into overdrive because Arsenal fans have been taking the piss out of his more ludicrous comments over the last 12 months. He's desperate to be right and is gambling big on us collapsing so he can say I told you so - because the alternative means he looks like the biggest mug on TV.