Is an oath valid if the wife commits to cleanliness in the coming period, given that the husband said to his wife: "You are to me as the back of my mother if you do not commit to cleanliness in the house, and I will send you back to an apartment in Faysal"?

Dhihar is valid whether immediate or conditional. If it is conditional upon an event, it does not occur until that event takes place. So, if the wife commits to keeping the house clean as the husband intended, Dhihar does not occur. However, if she violates this and does not commit, Dhihar occurs.