Is it permissible for a rabbi in a cheder to teach Torah during mourning

Is it permissible for a rabbi in a cheder to go teach children during the seven days of mourning?

If, due to the absence of the regular teacher, the children would not learn properly even with a substitute, then the teacher should go and teach them. However, if they are familiar with the substitute and will learn properly from him, the mourning teacher is prohibited from going to teach them.

It is written in Bedek HaBayit at the end of Siman 384: "It is written in Orchot Chaim that teachers of children have the same status as those needed by the public, and furthermore, it is a work that is lost when the children are idle." The Shach also wrote in SK 2 that it is permitted even from the first day. In Taz there in SK 1 it is written: "In TAU Nativ 28 it is written that I saw my teachers teaching halacha after three days, and from this, it seems to me that a teacher can also teach halacha to his students after three days, as this falls under the category of 'many need him,' since the breath of children in the cheder is more precious than the learning of adults, as it is said in Tractate Shabbat: 'The breath that has sin is not comparable to the one that does not.'" This means it is only permitted after the first three days. In Aruch HaShulchan it is written: "And a teacher of children, if there is no other in his place, it is obvious that it is forbidden to cancel the studies of the children in the cheder, and he should teach them even on the first day [Shach SK 2], and there are those who permit after three days [Taz SK 1], and one who relies on the first opinion does not lose." In Gilyon Maharsha it is written: "It is written in Teshuvot Rashal [at the end of Siman 66] that even if there is another rabbi, they will not study as diligently before a rabbi who is not permanent for them."