In all Cases affecting Ambassadors, other public Ministers and Consuls, and those in which a State shall be Party, the supreme Court shall have original Jurisdiction. In all the other Cases before mentioned, the supreme Court shall have appellate Jurisdiction, both as to Law and Fact, with such Exceptions, and under such Regulations as the Congress shall make.
Congress absolutely has, per the Constitution, clear authority to regulate the SCOTUS. How's this for an exception. The Supreme Court shall have no Jurisdiction in any case where any member of the court has a conflict of interest or even the appearance of a conflict of interest and refuses to recuse themselves?