Who is obligated to pay Zakat al-Fitr for the wife of a slave?

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The later scholars of our school stated that the obligation of paying Zakat al-Fitr for a slave's wife depends on her status: if she is a free woman, the obligation rests on herself; if she is a slave woman, the obligation rests upon her master. However, according to my understanding derived from the principles of the Madhhab, the obligation falls upon the master of the slave because he is obligated to provide for her maintenance (Nafaqah). This is analogous to the requirement for him to pay Zakat al-Fitr for his wife's servant, whom he does not own, due to the obligation of maintenance. Evidence dictates this based on the Prophetic saying: 'Pay the Zakat al-Fitr on behalf of those whom you support.' She is included among those supported. Furthermore, it is established by our scholars that if a person voluntarily undertakes the support of another, the obligation of their Zakat al-Fitr falls upon that supporter; thus, for someone upon whom maintenance is already obligatory, the responsibility is even stronger.