Derek John Bromley and an accomplice, John Karpany, were jailed for life for the 1984 murder of Stephen Docoza after his body was found floating in Adelaide's River Torrens.
His appeal is based on new psychiatric and psychological evidence which his lawyers argue makes accounts given by the prosecution's central witness Gary Carter unreliable.
Mr Carter suffered from a schizoaffective disorder which Bromley's lawyers say meant there was an "extremely high likelihood" that his account of events was inaccurate.
Bromley's brother's former wife Robyn Milera, who has campaigned for his freedom for more than 20 years, attended court on behalf of the family who were not able to travel to Canberra.
She spoke to Bromley on Tuesday morning and said he was feeling confident but also realistic, considering appeals processes have failed in the past.
"There's been many setbacks along the way, so we're prepared for anything," Ms Milera told reporters on Wednesday.
"But I do have confidence that the High Court will apply the law.
"Derek (told) me that he knows exactly what it is to be disappointed, to fail and to have to get back up and keep going."
In 2018, three SA Court of Criminal Appeal justices dismissed Bromley's application to appeal after ruling the new evidence about the key witness was not compelling.
But in its application to the High Court, Bromley's lawyers argue the lower court conducted a "flawed analysis" of the additional expert evidence and that other witnesses had not put Bromley at the scene of the offence.
"By taking the approach that it did, the Court of Appeal overstated the probative effect of what it said was evidence corroborating Carter," the appeal application said.
Ms Milera said the appeal was a critical opportunity for the family to heal from the trauma of Bromley's conviction.
"Derek's made a stand that, when you analyse it and think about it, is absolutely extraordinary," she said.
"He's not for a moment given up or stopped believing that eventually he would get a good result.
"He's said, 'I know I didn't do it, but I didn't get a fair trial', and that's what the legal question is."
Bromley's lawyers argue there was no independent corroboration of the prosecution's only key piece of evidence.
"That is, there is no independent corroboration of Carter's evidence that the applicant was involved in a violent altercation with the deceased, Mr Docoza. And that is the critical issue on this application," they said.
"The Crown fails to grapple with this fact. In the circumstances, special leave ought to be granted to appeal and the appeal allowed."
Five High Court justices are hearing the matter over two days.
Bromley has been in prison since 1985, having been denied parole because he has maintained his innocence.
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