The Privy Council in London on Thursday finished hearing an appeal by convicted double murderer John Barlow and is expected to make a ruling in the next few weeks.
Barlow was convicted in 1994 for killing Wellington father and son, Euguene and Gene and Thomas, after two previous trials ended in hung juries.
Five British law lords sitting in the Privy Council have heard arguments from Barlow's lawyer Greg King and deputy Solicitor-General John Pike representing the Crown.
Mr King spoke for two hours to outline why his client's conviction should be quashed outright or the case referred back to the Court of Appeal in New Zealand for consideration of new evidence.
The fresh evidence presented by Barlow's team relates to a report which discredits a technique to examine and match bullets.
The legal team said evidence supplied by an FBI agent in Barlow's third trial, which matched bullets found at the murder scene with some he discarded at a landfill site, relied heavily on this technique.
Mr King said the testimony of the agent was key in gaining a conviction, after the juries in two earlier trials were unable to reach a verdict.
Barlow's lawyers said the fresh evidence they presented cast doubt on his guilt and have called for his conviction to be quashed.
However, lawyers representing the Crown argued that the discredited matching technique did not apply to the type of bullets found at the scene and that other evidence existed to link Barlow murder of the two men in their office.
Deputy Solicitor-General John Pike said submissions about the bullet-testing technique relate to American tests, while the bullets used in the killing were European.
"Our case was that you can't necessarily apply the American problems, which are many, to the European scene, and the question is - did it matter in the context of the evidence anyway."
Greg King says he is pleased the British law lords allowed the appeal to be heard.
"All I can say is they gave us our day in court and that's what we wanted. That's what we've been fighting for for the last four years to get.
"It's disappointing for us that we had to come all this way to get that, but we're extremely grateful that the lords were prepared to hear our case and to analyse it in the way that they've done."
Barlow has always maintained his innocence of the crime.