Is it considered a sin to keep one's first wife under their responsibility without cohabitation, in response to the children's request, and what is the jurisprudential ruling in this case?

The wife has a right to lodging and marital intimacy. She is entitled to seek divorce if the husband abstains from fulfilling this right. However, if the wife agrees to waive her right and remain married, the husband is not obliged to divorce her. Muslim narrated from Aisha that Sawda bint Zam'a gifted her day to Aisha, which permits a woman to waive her right to equal division (of time). But if the wife is not content with her situation and does not seek divorce, then the husband must address the issue, grant her rights, and treat her kindly.