Well, I guess the answer is it wasn’t resolved.
The Bishop of Jerusalem also claimed primacy at one time or another, but things came to a head in 1051 when Leo IX (Rome) tried to interfere with churches who recognized Michael I Cerularius (Constantinople). The two bishops exchanged letters excommunicating each other and the Church split between Roman Catholic and Eastern Orthodox.
And obviously in 1517 the Reformation split the Church again over this and other issues. Ironically, this was not an original issue, but rather one that Luther investigated after the Pope’s refusal of support over the selling of indulgences.
In October 2007 the RCC and the EOC agreed to the primacy of the Pope as the first among bishops (primus inter pares) but the EOC rejected any increase in his authority.
Eastern Patriarchs acknowledged the primacy of Rome and Peter before the split.
St. John Chrysostom, Patriarch of Constantinople (c. 387):
(Peter), the foundation of the Church, the Coryphaeus of the choir of the Apostles, the vehement lover of Christ …he who ran throughout the whole world, who fished the whole world; this holy Coryphaeus of the blessed choir; the ardent disciple, who was entrusted with the keys of heaven, who received the spiritual revelation. Peter, the mouth of all Apostles, the head of that company, the ruler of the whole world. (De Eleemos, iii. 4; Hom. de decem mille tal. 3)
John Cassian, Monk (c. 430):
That great man, the disciple of disciples, that master among masters, who wielding the government of the Roman Church possessed the principle authority in faith and in priesthood. Tell us, therefore, we beg of you, Peter, prince of Apostles, tell us how the Churches must believe in God (Cassian, Contra Nestorium, III, 12, CSEL, vol. 17, p. 276).
St. Sophronius, Patriarch of Jerusalem (c. 638):
Teaching us all orthodoxy and destroying all heresy and driving it away from the God-protected halls of our holy Catholic Church. And together with these inspired syllables and characters, I accept all his (the pope’s) letters and teachings as proceeding from the mouth of Peter the Coryphaeus, and I kiss them and salute them and embrace them with all my soul … I recognize the latter as definitions of Peter and the former as those of Mark, and besides, all the heaven-taught teachings of all the chosen mystagogues of our Catholic Church. (Sophronius, Mansi, xi. 461)
Transverse quickly all the world from one end to the other until you come to the Apostolic See (Rome), where are the foundations of the orthodox doctrine. Make clearly known to the most holy personages of that throne the questions agitated among us. Cease not to pray and to beg them until their apostolic and Divine wisdom shall have pronounced the victorious judgement and destroyed from the foundation …the new heresy. (Sophronius,[quoted by Bishop Stephen of Dora to Pope Martin I at the Lateran Council]
, Mansi, x., 893)