Zilart and Promathia's storylines aren't really tied together much. Slightly linked, yeah, hence why when you first enter the audience chamber during the CoP missions the duke is "Out." There are some allusions to Zilart during Promathia's story, but nothing is really named.
Hell, the Nation storylines allude to Zilart as well, when you go to Muckvix's shop for the Yagudo Torch, he mentions "those two." But again, Nation storylines are seperate as well, aside from Zilart branching off of it.
Chains of Promathia is independant, and since Square already stated that you do not need to have progressed in Rise of the Zilart and Chains of Promathia to partake in Treasures of Aht'Urghan, we can assume it's independant as well.
Congrats on completing Zilart, so many people option not to, since there wasn't much to gain from doing the last fight until now. Just wanted to throw it out there that Aht'Urghan will most likely be stand alone, so you don't have to rush yourself through Promathia.