Wife asking husband’s permission to fast when he has another wife

Is it obligatory for a wife to ask her husband’s permission to fast if that is on the other wife’s day?

Praise be to Allah, and blessings and peace be upon the Messenger of Allah:

If the husband is present with her then she should not fast without his permission, because he may need her. But if he is with his other wife and it is the other wife’s day, then it seems that it is o.k. for her to fast with or without his permission.