Hallel on Hanukkah in the House of Mourning

Mourners sitting shiva during Hanukkah, should they recite Hallel in the morning prayer at the house of mourning?

According to the Mishnah Berurah, on Hanukkah and Purim, Hallel is not recited, and afterwards, the congregants complete it at home. The custom follows the Aruch HaShulchan, where the mourner recites Hallel with them on Hanukkah and Purim but does not lead the prayers during Hallel.

Written in the Mishnah Berurah, Siman 131, S"K 20: "Hallel is not recited in the house of mourning. On Rosh Chodesh, when praying in the house of mourning, even if they go home afterwards, it is not necessary to recite it. On Hanukkah, when the Hallel is completed, it must be recited later at home. However, on Shabbat that coincides with Rosh Chodesh, or if the deceased was buried on a festival, even if praying in the house of mourning, Hallel is recited because there is no mourning." Also written in Kitzur Shulchan Aruch, Siman 227, Seif 6. And in Shulchan Aruch HaRav, Orach Chaim, Siman 131, Seif 4: "What is being discussed? The Hallel of Rosh Chodesh, which is obligatory only by custom and is not recited in the house of mourning. But on Hanukkah, when Hallel is obligatory by the decree of the sages, it is recited even in the house of mourning, as the mourner is also obligated. However, it is customary that the mourner for 30 days, and for a parent for 12 months, does not lead the prayers during Hallel, as the congregation is in joy at that time, similar to Shabbat and festivals."