In a detailed and cogent article on Obsidian Wings, Hilzoy makes a strong case that the so-called "nuclear option" of eliminating judicial filibusters may itself violate the rules of the Senate. I am not an expert in these matters, and don't even know where to begin to evaluate whether the argument is correct, but the jist of it is that the same Senate rules which require a 3/5 majority (60 votes) to end a filibuster also require a 2/3 majority (67 votes) to end a filibuster of a change in the Senate rules. As a result, the only way to invoke the "nuclear option" (which involves a change to Senate rules) is through parliamentary maneuvering to prevent any debate on the rule change at all.

Whether this interpretation of Senate rules is correct or not, there's a persuasive case in there that invoking the "nuclear option" would either break or bypass Senate rules. In this scenario, it is very plausible that you could have one side recognizing one set of Senate rules (rules which prohibit judicial filibusters), while the other side recognizes a different set (the rules as they are today unchanged) claiming that the change to the rules was improper and therefore invalid.

The Constitution gives the Senate the power to make its own rules, but does not specify what process would be used to adjudicate a dispute over the mechanism used to change the rules. A lawsuit is a highly probable outcome, which would put the courts (probably the Supreme Court) in the odd position of having to decide an internal dispute in the Senate.

There is likely to be some question as to whether the Supreme Court even has the authority to rule on such a case, since the Constitution only recognizes the Senate's power to make its own rules. The Supreme Court could very well decline to hear the case on the theory that they have no jurisdiction (and that the Senators are supposed to be grown-ups anyway). It is extremely unlikely that the Senate itself could resolve such a dispute in a manner which all sides agree is fair. Thus you could be left with two disputed sets of Senate rules, and no way to resolve the issue.

At this point, the speculation starts to go off into lala land.

You could get a situation (for example) where the Senate votes to confirm a judicial nominee under the new set of rules, but the minority doesn't recognize the confirmation because under the old rules the nominee would have never been brought to a vote. Who then decides if the judicial nominee takes a seat on the court or not? (Presumably the judiciary itself can rule whether a nominee has been properly confirmed--after all, congress gets to decide if its own members were properly elected).

Things get even weirder if the rule change is extended to eliminating all filibusters. You could have laws which pass under one set of rules but not others, leaving situations where the courts have to decide not just hot to apply the law, but what the law actually is.

This whole process is very much like watching the Senate getting sucked into a vortex of hyper-politicization. None of the Senators seem to really want to go in this direction, but external forces (particularly the extreme activists on both sides) won't let them stop. When I see things like this "not one dime" campaign, it truly saddens me to see how far we've already gone down the road of zero-compromise, zero-tolerance politics ruled by the most extreme elements. This is not good for Democracy, and it is not good for our country.