Three appeal court judges have reserved judgement in the case of the murder conviction imposed on Ruth Ellis, the last woman to be executed in Britain.
The jury took 14 minutes to convict Ruth Ellis
Ellis's family want the verdict reduced to manslaughter and say she suffered from "battered woman syndrome".
In 1955 she killed her lover, racing driver David Blakely, in Hampstead, north London.
On Wednesday the Court of Appeal declined to hear evidence from a medical expert who aimed to prove that nightclub hostess Ellis was suffering from the syndrome.
Lord Justice Kay said it was not admissible in law, although this did not mean the court regarded it as wrong.
Appeal judge Lord Justice Kay, sitting with Mr Justice Silber and Mr Justice
Leveson, also said it was very unusual Ellis had not appealed and seemed "resigned to her fate".
In a statement not produced at the original trial, Ellis said she had intended to kill herself at the scene, but ran out of ammunition.
Michael Mansfield QC said Ellis had received advice from her counsel that there were no grounds for appeal but this had been wrong.
He said her lawyers, and everyone involved in the trial - including the judge, Mr Justice Havers - had misunderstood the law on the defence of provocation.
Counsel for the Crown, David Perry said Ellis had calmly carried out a "planned act of revenge motivated by jealousy".
Provocation could only be used as a defence in the case of a sudden
loss of self-control in response to a provocative act, he said.
On the day of the shooting, Mr Blakely had not been violent, but had broken off their affair, the judges were told.
Ellis's sister is leading the family's fight to get the verdict overturned
Even if that amounted to provocation, her response to it was wholly
disproportionate, said Mr Perry.
At her Old Bailey trial, Ellis said: "It was obvious that when I shot him I intended to kill him," after which the judge ruled out provocation as a defence.
But an eyewitness had described Ellis as "wild-eyed and talking abstractedly" just before the shooting.
Speaking outside the court, Ellis's son-in-law Mike Blackburn described the effect Ellis's execution had had on her daughter, his wife Georgie.
Reminders of execution
She was three when her mother was executed and was adoped three years later when her father hanged himself.
Mr Blackburn from Beverley, East Yorkshire, said: "She was constantly reminded of it. Complete strangers would come up to her and ask her if she was Ruth's daughter.
"It haunted her all her days."
She campaigned for the case to be heard at the Court of Appeal but died of cancer in 2001.
"It's just so tragic she's died before it's been completed," he said.