If the election results are in doubt, the incumbent president should stay in office.
No he shouldn't By law, the term of office ends at noon on January 20, 2021. If the election is still in dispute, the Speaker of the House becomes President. It is not clear if the term of office then expires when the election is no longer in dispute or at noon on January 20, 2025. In my opinion, if we get to this point the election will never be not in dispute.
The other thing that could happen is the state legislatures use their plenary authority under Article 2, section 1B to appoint electors as they see fit; skipping the election entirely.