Is a person considered to be among the Prophet's (PBUH) Household because their grandmother is a descendant of Al-Hasan ibn Ali, and do the People of the Household have a special intercession from the Prophet (PBUH) on the Day of Resurrection?

"Your mother being a descendant of Al-Hasan ibn Ali adds to your merit and goodness, but it does not make you a part of the Ahl al-Bayt (Household of the Prophet); because lineage is only traced through fathers. As for the special intercession of the Prophet, peace be upon him, for the Ahl al-Bayt, this has not been mentioned among the known types of intercession, which include: relief from the horrors of the Standing (Day of Judgment) and hastening of the reckoning, admitting some people to Paradise without accountability, intercession for those who deserved Hellfire, intercession for sinners who entered Hellfire to be brought out of it, and increasing the ranks in Paradise for its inhabitants."