In its judgement released today, the Supreme Court found the criteria for leave were not met, saying it was "not persuaded that the Court of Appeal's assessment in relation to either of the issues now raised can be faulted".
"Neither singly nor in combination do they raise a concern that there may have been a substantial miscarriage of justice," said the report.
Last weekend, members of Augustine's family joined a gathering of about 120 people representing 50 families affected by some of the country's most shocking and high-profile murders.
The group met at the Dunes Hotel on Waiheke Island to share their harrowing stories for the first time.
At that meeting, Augustine's father Charlie said Davis' attempts to appeal his conviction continued to put the family through "an emotional rollercoaster ride", the Herald on Sunday reported.
"My son was murdered on the day the All Blacks played France in their first game of the last World Cup. So this World Cup was a sort of semi-happy time because we won it," he said.