Perhaps it was fate, the the baron actually doesn't have any true progeny since both his children weren't of his blood.
If he were to die his entire family will die with him but blood doesn't matter for what matters is the name of the family or the person, the name is remembered by everyone as it won't be forgotten especially if the person is important.
The daughter of the Baron, Ari Barkemy was not a true Barkemy as she was the last remaining child of the Baron.
Especially since her brother and the Baron's son was killed by the church for an sacrifice which was simply done to threaten the Baron as his son was his proclaimed heir leaving only his daughter which will naturally end his whole family and his rule on the City.
Will it be the end of the house of Barkemy? One can truly wonder what the future holds as it's a mystery after all.