Is it permissible to shave a part of the mustache where no hair grows to avoid embarrassment, by analogy with the permissibility of trimming it to treat a wound or the like?

The preponderant opinion is that it is impermissible for the one intending to offer a sacrifice (udhiyah) to remove anything from his hair, nails, or skin, based on the Prophet's (peace be upon him) saying: "When you see the crescent of Dhul-Hijjah, and one of you wishes to offer a sacrifice, he must refrain from [cutting] his hair and nails." However, it is permissible for the one offering a sacrifice to remove something from his hair, nails, or skin due to an excuse and to avoid hardship, such as if refraining from doing so would cause him harm or embarrassment, or if he has a wound and needs to cut the hair around it.