But a third wanted ongoing name suppression and his application was heard before Judge Claire Ryan.
The man indicated he would plead not guilty to corruption and bribery charges.
He argued publication of his name would cause extreme financial hardship to his family. He also cited health and employment concerns.
Judge Ryan said more information about the health conditions should have been submitted.
“It’s just claims by him,” the judge said. “He makes a sweeping statement.
“It’s not great, to be honest. This matter’s been around since December,” Judge Ryan added.
She said suggestions of stress were also not enough to grant name suppression.
“Every human being has a level of stress,” the judge added. “I would have expected a report from [his] GP.”
Judge Ryan added: “Many people have their names before the public for very serious offending. They still have a fair trial and the jury reaches a fair verdict.”
She said the case related to an allegedly “deceptive and corrupt scheme” where men involved sought to unfairly secure the award of lucrative contracts.
The alleged victims included Regional Facilities Auckland (RFA) and the former Counties Manukau District Health Board.
RFA made payments to one of the accused men totalling $657,107.31, according to court documents.
One defendant allegedly embedded another man at the health board in a fraud worth $85,370.25.
More alleged frauds involved payments of $106,556.99, $78,530.63, $4025, and $34,270.
The SFO has claimed the men conspired to deceive those public organisations and funnel money into a joint business for personal benefit.
“In this case the alleged offending is clearly serious,” Judge Ryan added.
She said people were entitled to know “New Zealand’s reputation for anti-corruption is being protected”.
The Herald opposed the name suppression application and the SFO also opposed it.
Judge Ryan dismissed the name suppression application but defence counsel Jeremy Bioletti asked if he could appeal.
The defence now had ten working days to send an appeal to the High Court.
The court also heard one of the defendants had asked for more time before entering a plea.
John Weekes, online business editor, has covered courts in New Zealand and Queensland, as well as crime, politics, breaking news and consumer affairs.