Lundy has previously been convicted of the murders of wife Christine 38, and her 7-year-old daughter Amber at their Palmerston North home in August 2000.
Lundy's lawyers went to the highest court in New Zealand on Friday to seek leave for yet another appeal on the infamous case.
They argued the Court of Appeal was wrong to uphold the convictions despite finding the evidence around the brain tissue was inadmissible.
Christine and Amber Lundy were found dead in their Palmerston North home, having been hacked to death, likely with an axe or tomahawk. The murder weapon was never found.
On the night of their murders Lundy had checked into a motel in Petone where he called an escort about 11.30pm.
It was fiercely contested during his first trial whether or not it was physically possible for Lundy to have travelled between Wellington and his family home in a time that would have allowed him to be in the house at the time of the murders.
Lundy was first convicted of their murders in 2002, and his first appeal attempt resulted in the court increasing the non-parole period of his life sentence to 20 years.
In a 2015 retrial at the High Court in Wellington, Lundy was again convicted of the murders.
He appealed to the Court of Appeal last year, which found the evidence around the brain tissue should not have been presented to the jury, but decided to uphold the convictions anyway.
The Crown argued Christine Lundy's brain tissue was found on the polo shirt her husband wore on the night of the murders.
Tiny spots consistent with dried blood were also found on the shirt which were found to contain Amber's DNA.
At the heart of the defence appeal last year was that the jury at the 2015 retrial should never have heard the mRNA evidence, as the analysis was too new and unproven to be relied upon.
The mRNA evidence, which is similar to DNA evidence, was used to rule if the tissue on Lundy's shirt had a human or non-human source.
The Court of Appeal ruled the evidence was inadmissible, but believed Lundy would have been found guilty regardless.
In the Supreme Court last week, Lundy's lawyer Jonathan Eaton QC said the Court of Appeal's finding should have lead to them overturning the convictions.
He also raised concerns about the trial judge not giving the jury a demeanour direction, and about the admission of the IHC evidence at the time.
In a decision released today, the Supreme Court granted leave to appeal on the first point, but not on the issues of the IHC evidence or the demeanour direction.
"We see no appearance of a miscarriage in the way the Court of Appeal addressed the issue in relation to the applicant's demeanour at the funeral of the deceased," the decision said.
The absent demeanour direction refers to Lundy's much-publicised funeral behaviour.
Images were published of Lundy staggering and being supported by the arms at the funeral of Christine and Amber.
Eaton earlier said the scene had "become folklore in New Zealand, was referenced in the defence opening statement and was the elephant in the room at Lundy's second trial.
"It needed to be addressed by specific [judicial] direction in order to avoid the risk of the jury engaging in demeanour reasoning."
The IHC evidence is what allowed the Crown to show the spots found on Lundy's shirt were from the central nervous system - brain or spinal matter.
