The description of the Blackstaff magic item in the Waterdeep: Dragon Heist adventure (p. 190) states:
When the Blackstaff dies, the spirit of that individual becomes trapped in the staff along with the spirits of the previous Blackstaffs.
If a lich is worthy enough to attune to the staff, then when it dies, does the Blackstaff have precedent over the lich's phylactery?