The story given is that Rock was legit pushing for Cody to turn and join. HHH and Cody didn't want to do it, but ultimately Nick Khan deferred to Rock. HHH and Cody still thought it was a bad idea, and so HHH basically told Cody to just refuse.
They then approached Cena about it and he was fine with it. This sounds believable to me. Cody turning would've been a waste of their most over pure babyface in a long time and the guy they ended Roman's title reign for.
I get why Rock would want to do it, it would've put the Final Boss over even more and if he wrestled Cody or Roman WWE is not gonna want Rock to go over.