The Iranian Monarchy was overthrown following the Islamic Revolution in 1979 with the Šâh Mohammadrezâ Pahlavi going into exile.
Under the Pahlavi Dynasty the law of succession stated the Šâh must profess the Islamic faith, his mother must be an Iranian citizen, a Muslim and not descended from the previous Qâjâr dynasty which rules out the sons of Reza Shah by his 3th (Turân Amirsoleymâni) and 4th (Esmat Doulatšâhi) wives and their male line descendants.
Except Mohammadrezâ Šâh, only Prince Alirezâ Pahlavi was eligible among the sons of Rezâ Šâh. With the death of Prince Alirezâ Pahlavi, only his son is eligible. But the mother of his son Prince Patrick Ali Pahlavi (nephew of Mohammadrezâ Šâh Pahlavi) is not a natural born Iranian citizen, nor the Iranian parliament had approved this legal term Irâniyol Asl [ایرانی الاصل] for her (as opposed to Princess Fawzia Fuad of Egypt, first wife of Mohammadrezâ Šâh). As a result Prince Patrick Ali Pahlavi is NOT eligible.
Moreover, Prince Patrick Ali Pahlavi also married a non-Iranian woman and his sons face the same issue, even if an exception is made for Prince Patrick Ali Pahlavi. Thus, Prince Reza Pahlavi is the only remaining eligible person. With his death, nobody can claim the throne even theoretically.
Given that the modern claim calls for a new constitution, and thus the bylaws need not necessarily be in accordance with the previous constitution in the event the monarchy is restored new succession rules may be established like the abolition of primogeniture and thus giving the daughters of the current Crown prince Rezâ Pahlavi a legitimate claim to the Peacock Throne.