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In November last year, the Supreme Court ruled that internet entrepreneur Dotcom and his three co-accused could be extradited to the United States to face criminal copyright charges - but also allowed them to challenge the decision through a judicial review.
The hearing will be heard in the Court of Appeal on Thursday.
However, Dotcom had argued the three judges should recuse themselves as they were the same panel that dealt with the appeals in the first place.
When they refused, he asked the Supreme Court for leave to appeal their decision refusing to recuse themselves - on the grounds they had not provided their reasons - and to put the main appeal on hold until that matter was resolved.
However, in a written decision, the Supreme Court dismissed the application.
The judges said if Dotcom wanted to raise concerns about the composition of the Court of Appeal panel, he could do that in an appeal against the substantive decision when it is made.
"While that involves some potential for duplication of effort if an appeal against the Court of Appeal panel's decision not to recuse itself were ultimately successful, that has to be balanced against the disruption to the processes of the Court of Appeal (and the inconvenience to the other parties to the remitted appeals) if this Court made the order the applicant seeks."
The judges said Dotcom had already indicated that if the court did not make the order sought, he would pursue his application for leave to appeal, and he was entitled to do so.
Dotcom is fighting extradition to the US on copyright charges along with three co-accused.