If you use the search box and type in Georgia, you will find a selection of articles on the dynastic issues regarding the Georgian monarchy, and the "battle" between the two branches, briefly united in marriage.
According to a source close to Prince Nuzgar Bagration-Gruzinsky's Chancellery, Prince Giorgi is now considered a royal dynast and will inherit the de jure authority of his grandfather. Under Georgian dynastic rules, Giorgi did not have a royal claim given his father’s actions. However, it’s my understanding that HRH Crown Prince Nugzar received a request from Ilia II, the Patriarch of Georgia, to grant Giorgi full dynastic rights a year or so ago. He consented to the request, and little Giorgi is now a Gruzinski dynast."
Prince Nuzgar's older daughter, Anna, who is divorced from Prince David Bagration, lives in Georgia with her two daughters from her first marriage, and Prince Giorgi. Prince David has visitation rights.
Prince Giorgi was born September 27, 2011.