Every SSPX bishop has at one point speculated that they MIGHT be illegitimate. That already makes them DOUBTFUL POPES and on that account NO POPES (Papa Dubius Nullus Papa) ... for all intents and purposes.
None of the popes since the council can be considered "doubtful," according to the teaching of the canonists.
Wernz-Vidal explains that the phrase "Papa dubius est papa nullus" can be understood negatively or positively. It is understood positively
when, "after a careful examination of the fact, competent men in the Catholic Church would pronounce: 'The validity of the canonical election of this Roman pontiff is uncertain'." This has not taken place.
Werns-Vidal go on to explain that the words "no pope" do not
apply to a pope who was at first accepted by the entire Church, "but concerning whose election so many difficulties are subsequently
brought to light that he becomes 'a doubtful pope' in such a way that he would thereby forfeit the pontifical power already obtained." Once his election is accepted by the entire Church, as has been the case with every pope from John XXIII until now, any later doubts that arise, even if the doubts are real, will not make, nor will it suggest, that he is "no pope". The reason is because the acceptance of the election by the entire Church makes his legitimacy infallibly certain, and he will legally retain the pontificate until he ether dies, resigns, or is declared deposed by the competent authorities due to insanity or heresy.
way in which a pope can be legitimately deemed doubtful is if his election was always in doubt from the beginning
and continues to remain in doubt. "But the other part of this axiom could have the meaning that a Roman pontiff whose canonical election is uncertain and remains subject to positive and solid doubts after studious examination, absolutely never did acquire also the papal jurisdiction from Christ the Lord." In this case it would be legitimate to refuse him obedience, but not because of doubts that arise after his election has been accepted as pope by the entire Church.
Obedience can be refused to a legitimate Pope if the command given is illicit, being opposed to faith or charity, but he must be obeyed in all other things. I would also note that, whereas I fall into the category of "Recognize and Resist", I have never refused any direct command of the Pope. Maybe we should come up with a third category to further divide traditional Catholics.