Is a single Tayammum sufficient for both the face and the hands if both areas are wounded during Wudu?

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If a person washes their sound face first and then performs a single Tayammum for both the face and the hands, this act is insufficient because it results in neglecting the obligation (fard) for a part of the face and a part of the hands simultaneously. This is differentiated from the case of Tayammum for the entirety of ritual impurity, where a single act suffices for all limbs at once, because in the latter case, the ruling pertains to the totality, whereas here, the act only substitutes for specific parts.