That's because it was traditionally mongolian. The dzangars that were wiped out by kangxi were a mongol tribe. There are still plenty of Mongols there, that's why we don't like those foreign funded terrorists either.
I'm not arguing against it. I'm pointing to the irony how China is expected to give up historical claims yet everyone else is still making themselves. Uighurs are not going to give up to Mongolians. If it's Mongolian territory, Uighurs have no claim. Mongol history is known for conquering. One really can't stake claim when that territory was conquered from them.