New Zealand's Supreme Court will release its decision tomorrow morning on whether the man who murdered Grace Millane will be publicly identified.

The country's top court emailed the killer's legal team, Crown, and media at 2.35pm today informing them its judgment will be delivered at 11am on Tuesday.

The killer's identity was due to be revealed last Friday morning, as ordered by the Court of Appeal, but with just minutes before the deadline, the Supreme Court intervened and kept decided to keep suppression in place until it made a decision.

Chief Justice Helen Winkelmann, and Justices William Young and Mark O'Regan gave the killer's legal team until 5pm last Friday to file their arguments, while the Crown and media were ordered to do the same by 10am today.

If the Supreme Court's rules to lift the order, it will end a suppression the killer has maintained since his first court appearance after killing the British backpacker in December 2018.

Last week, the Court of Appeal also delivered its decision on the killer's appeal of his conviction and sentence for fatally strangling Millane, whom he met on the dating app Tinder.

Both appeals were dismissed.

Grace Millane's killer continues to have suppression. Photo / Michael Craig

However, the man continues to claim his innocence and asserts Millane died during rough sex. He has already said a second appeal to the Supreme Court will occur.

When considering the killer's sentence, the Court of Appeal said Justice Moore was correct when imposing a non-parole period of 17 years.

"To be plain about matters, it really is very difficult to imagine much greater vulnerability than the situation Ms Millane found herself in on the evening of Saturday, December 1, 2018."

After Millane died, the killer also searched the internet for methods of how to dispose of her body - which was discovered in a grave in the Waitākere Ranges - looked at pornography online, and took intimate photos of her naked body.

He also went on another date while Millane's body remained in his downtown Auckland hotel apartment.

"We conclude that this behaviour is indicative of a degree of wholly self-regarding wickedness throughout the incident and its aftermath."

The court hearings have cost thousands of dollars in legal aid, with the Ministry of Justice revealing the murderer's bill was more than $400,000.

In a statement after the Court of Appeal's judgment, Millane's family thanked the judges and others involved with the case.

"She was enjoying the first of what would have been a lifetime of adventures before her life was so cruelly and brutally cut short by her murderer."

The Millane family said while the focus would inevitably be on the outcome of the legal process, they had their girl at the forefront of their minds.

"Grace, you are, and will always be, our sunshine," they said