They were first reported missing by family members before police found their bodies a few days later.
Martin Hone already pleaded guilty to the men's murders and was sentenced to life with a minimum non-parole period of 20 years.
Pulemoana entered an appeal to the Court of Appeal on January 31 but it was dismissed in a decision released on July 9.
Pulemoana's lawyer last week filed an application with the Supreme Court.
A court spokesman said the court was now awaiting submissions from both sides of the case and the court was yet to decide whether leave was granted for the appeal to go ahead.
Pulemoana stood trial with co-accused Mikaere James Hura, 21, who was found guilty of the manslaughter of both men and was jailed for six years.
Richard Te Kani pleaded guilty to manslaughter and is serving a sentence of 15 years in jail.
The Fleets' deaths came after Raymond Fleet became involved in a methamphetamine manufacturing enterprise which did not return the yield those who ordered it anticipated.
At the trial, and at Hone and Te Kani's sentencings, it was emphasised James Fleet had not taken part in manufacturing the drug but had been "in the wrong place at the wrong time" when he and his uncle were forced into the vehicle that took them to the place where they were to lose their lives.
James Fleet's mother, Bron Fleet, told the Rotorua Daily Post today she accepted Pulemoana had a legal right to appeal.
"I can do nothing but trust in our justice system to make the appropriate finding."
She said every day was a "challenge" to manage the grief of losing James and to keep their family strong and together to support each other.
"Every court hearing is like pulling the scab off a wound that's trying desperately to heal, it's devastatingly painful. I hope one day soon that this will finally be over and James can rest peacefully knowing that those who took his life are held accountable."
