The basic historical document regarding the Jinshin War was "Nihonshoki" (Chronicles of Japan), which does not say that Prince Otomo became the crown prince or was enthroned.
過去から現在まで壬申の乱の基本史料は『日本書紀』であり、これは大友皇子が皇太子になったとも、即位したとも記さない。