Is it permissible for a husband to live with his wife in the same house, while foregoing speech and intimacy, and sufficing with greetings and necessary conversation for a temporary period, for the sake of their infant daughter's welfare, knowing that he sees no chance of reconciling with her due to her disobedience, and how should her maintenance be handled in this case?

A wife is obligated to obey her husband and move with him to a safe country, unless she stipulated otherwise. If she refuses to move, she is considered disobedient (nashiz) and her right to financial support (nafaqah) is forfeited. It is permissible to keep the wife without her rights by mutual agreement for the sake of the children, as Sawdah did with the Prophet, peace and blessings be upon him. A disobedient wife loses her right to financial support and equal turns (qasm), but her right to sexual intercourse and enjoyment remains. If she agrees to waive this right and remain in a suspended state, there is no harm. If you do not wish to spend on her due to her disobedience or her consent, then you are only obligated to spend on your daughter. As for the rent for accommodation, she is not obligated to pay any of it, and the wife is responsible for her own expenses. We advise you to release her kindly and marry another woman who will help both of you maintain chastity.