Neut wants both of them dead. He's not going to give a shit which one dies first since either one of them dying means he's fulfilled 50% of his win-con.
Having the Emperor lynched means that he knows one of his targets AND still has a full player-list so he has more time to find the Pleb Leader or try and find the Guard first so that he's safe or some shit like that.
So I don't get you.