Three possibilities:
1) Francis freely resigns (albeit with public calls for him to do so). He can then be deposed/degraded as a bishop after a new Pope is elected (cleanest way is to re-elect Benedict, but he neither wants nor has strength for it). Risk of some claimimg an invalid resignation, but no more than was true for Benedict. Also, would a Pope-emeritus Francis be as quiet as Pope-emeritus Benedict?
2) Election declared invalid. Cardinals created since would be invalid, also likely for episcopal appointments (but episcopal consecrations would be both valid and licit, as the mandate for consecration enjoyed presumed validity at the time). Better chance of a good pope being elected, as many shady Cardinals are barred (invalid creation), aged out, or are marginalized due to involvement in the politicking or subsequent scandal (Wuerl, Maradiaga). Triple-difficulty of proving the allegations of politicking (unfortunately the recent scandals and Viganò’s letter have little probative value here), getting Francis to accept/forcing him to accept, then garnering sufficient public acceptance of the invalidity. Very high risk of schism, with no guarantee that the “good guys” end up holding the high ground.
3) God interjects a word and addresses the problem for us. If He is especially merciful, He also calls Benedict home immediately before the Conclave is to begin (removal of any doubt of validity for new Pope)... Added bonus if this comes at the end of the Pope’s intransigent refusal to accept proofs of #2, which would give justification to bar recently-created (invalid) Cardinals from the Conclave.
All of them are messy. Don’t worry... pray.
Censeo praedatorium gregem esse delendum.
4) Francis, with the secular media providing air cover, rides this out until natural death 8 years from now at age 90. His entire gang of creeps and crooks stay in power and is bolstered by even more creeps and crooks. Francis abolishes the entire catechism with respect to sexual morality, approving homosexual acts, homosexual marriage, birth control, fornication, etc... and throws in women deacons for good measure. The Church is the USA & Europe hemorrhages 50% of its members A new conclave convenes, and with 90% of its voting members being Francis-appointees, elects a liberal, openly gay American Cardinal as the 267th pontiff. The cardinal takes the name Francis II.