Associated with Nestorians by Westerns. But not Nestorian by dogma. The Persian kings took the chance of the schism to create their “own Christians” so they didn’t have to fear an alliance between “their Christians” and the Christian Roman Empire
As far as i know there were christians in iran (during Parthian rule & before) before Nestorians relocated to then Persia under the Sassanid rule, after theological disputes with the Roman empire. True later about allegiance to the Sassanid for political reasons vis a vis conflict with the Roman empire.
There are still theological disputes & divisions within the Assyrian Church of the East & its connection to Nestorianism. I know quite a few Assyrians who lived in iran; loads of them in iran during the shah’s era.