It is very interesting to note that history is always in some kind of cycle - repeating itself ad nauseum.
Iran is prepared to fill the supposed Iraq vacuum
It is no surprise that Iran would fill in any power vacuum real or imagined.
It is a natural progression of events.
The three parties Iraq, Turkey, and Iran also have a vested interest in reducing the Kurd influence. One of the reasons that Iran has to act (from its point of view).
Kurdistan would be an unmitigated disaster for Iran. And they would likely go to war to prevent that from happening.