:You are right! Maybe Altan's descendants wanted to advertise his glory. However, the Ming-shi also says the Oirat leader and Esen's grandson, Ish-Temur, controlled Qaraqorum in the 1460's and defeated Unebolad and Mulikhai. Bek Arslan failed to occupy the city after Ish-Temur's death. It is so pity that there are many different stories.