How can the father's desire to sever ties with his wife's family be reconciled with the wife's desire to maintain communication with her family, taking into consideration the husband's promises to his wife before marriage and their shared living arrangements?

The spouses are not obligated to obey their families in boycotting or shunning the family of the other party. In fact, it is not permissible to do so, based on the Prophet’s (PBUH) saying: "It is not lawful for a Muslim to abandon his brother for more than three days," and his saying: "Obedience is only in what is good (ma'ruf)." It is advisable to bring viewpoints closer and advise the families to overlook and forgive. If that is not possible, then each spouse should inform their family that they have no connection to the matter and do not wish for their relationship with their in-laws to be ruined.