The accused denies charges of importing, possessing and supplying methamphetamine. His case is due to go to trial later this year.
An interim name suppression order was first granted by Judge John Macdonald and extended at the second appearance by Judge Peter Rollo to protect the ongoing police investigation.
However, in November last year the man tried to have that name suppression order continued for his own reasons.
That application was two-fold, suggesting that he would face extreme hardship if named and that his fair trial rights would be jeopardised.
Judge Richard McIlraith declined the application, deeming publicity did not equate to extreme hardship and there was no evidence fair trial rights would be affected.
"There is nothing unusual about [the defendant's] situation," he said.
The man's lawyer, Greg Bradford, immediately launched an appeal against the decision.
That appeal was heard today in the High Court at Auckland, advanced on the same points again.
Bradford said at this stage: "Open justice needs to wait a little bit longer".
Crown prosecutor Jessica Pridgeon has said there was no evidence backing the application.
She said Parliament had made it clear that being well known does not mean the defendant will suffer extreme hardship.
Justice Rebecca Edwards has reserved her decision on the appeal.
