The woman charged with murdering two children whose bodies were found in abandoned suitcases is fighting to keep her name secret.
She is the mother of the children, and is due to stand trial next year after earlier pleading not guilty to two charges of murder.
The High Court refused to grant continued name suppression for the woman in March and her lawyer lodged an appeal, which has been heard in Auckland on 8 May.
Justice Anne Hinton declined name suppression because the threshold test of extreme hardship or endangering safety were not established.
"While it sounds harsh, that is the relevant test," she said in the decision.
Today the woman's lawyer Chris Wilkinson-Smith submitted that publication would result in undue hardship or risk her safety and that this risk could not be dismissed.
Both the Crown and the lawyer for Stuff, NZME, Discovery and RNZ, Tania Goatley, argued there was no evidence naming the woman would cause her undue risk or present a risk to her safety.
The Court of Appeal has reserved its decision.
Much of the appeal cannot be reported due to suppression orders.
Last week, the 42-year-old appeared briefly in the High Court in Auckland for an administrative hearing.
As she was led back to the cells, she raised her hand and said she was going to prove her innocence.
The bodies of her two children were discovered in suitcases last August, after a family bought the contents of an abandoned storage locker in Papatoetoe, in an online auction.
The family had transported the goods, including the suitcases, home to Clendon Park on a trailer before unpacking them.
At the time, police said the children had died a number of years before their remains were found.
Stuff reported the father of the children had died in 2017, a year before the mother is thought to have arrived in South Korea. The children were aged five and eight at the time of his death.
The woman, a Korean who is a New Zealand citizen, was taken into custody in South Korea last September after the Korean Ministry of Justice received a request from New Zealand for her provisional arrest in connection with the deaths of the children.
The Seoul High Court approved the extradition last November, and the woman was surrendered to three New Zealand police officers, who escorted her home.
She appeared at Manukau District Court less than 24 hours after landing in New Zealand.
The identities of the two children remain suppressed by the coroner.