Mulcair could end all the doubting by renouncing his French citizenship. Not only heading off a big future problem, but put paid to the persistent narrative within the NDP that he is over proud, does not listen, etc.
A lot of people would not like that, actaully be dissapointed in him. Many would see it as bowing to bullying. It would not be perceived that way at all if he were to do it early on. He can make his apologies, speak of it as a sacrifice he felt he needed to make, that he did not want it to be a distraction...