Mukhi and purvi are no more man and wife ,There marriage is now illegal and they are no more husband and wife on any legal grounds
Under section 108 of indian constituion, a person who has not been heard of for SEVEN or more than SEVEN years is declared dead by the court.
Sakshi is back after 5 years so her existence has make mukhi's second marriage null.
Proof links:
https://indiankanoon.org/doc/188287577/#:~:text=Though%20Section%20108%20of%20the,proved%20as%20any%20other%20fact.
Mukhi and purvi's marriage is no more valid. If they go by the section 108, Mukhi is only sakshi's husband and he need to complete all the legal formalities and need to divorce her at first place.
So belonging to this explanation, Purvi does not has any identity among the law and every relation related to her are also legally finished.
So matter is in sakshi's hand whether she will sign divorce papers or not.
There is long divorce battle ahead then only purvi and mukhi can do marriage again.