Avon coroner Paul Forrest found former tiler Mr Hogan had "unlawfully killed" the six-year-old as a holiday attempt in Crete to salvage his marriage went wrong.

But Sir Anthony May, sitting with Mrs Justice Dobbs in London yesterday, said of Mr Forrest's conducting of the inquest: "The question of (Mr Hogan's) mental state was simply not addressed."

The judges now have to decide whether to overturn the verdict delivered at King's Weston House last March and order a fresh inquest, or possibly substitute their own verdict.

Sir Anthony said the court hoped to give its judgment next week.

Mentally unstable Hogan, 34, from Bradley Stoke, near Bristol, pushed Liam and his two-year-old sister, Mia, off a balcony before jumping himself following a row with his then wife, Natasha.

Liam died, but Mia survived the 50ft plunge from the fourth-floor balcony of the Petra Mare Hotel at Ierapetra, Crete, in August 2006.

A Greek court found Mr Hogan not guilty of murder but ordered him to be detained in a psychiatric unit. The jury decided he had been suffering from "an earthquake of insanity". His older sister, Christine O'Connor, of St George, Bristol, yesterday asked the High Court in London to overturn the unlawful killing verdict on the grounds that Mr Hogan was "not in control of his actions" because of his psychotic state.

The Hogans had gone on holiday in a "make or break" attempt to patch up their failing marriage and the incident occurred shortly before they were due to return home.

Before the balcony plunge, an argument had started between the couple and the then Mrs Hogan said she intended to leave her husband and take the children with her.

The court heard the Director of Public Prosecutions had indicated there would be no prosecution of Mr Hogan in this country for murder.

The former wife, Natasha Visser, has since remarried and now lives in Australia.

Yesterday, her counsel Gareth Patterson argued a verdict of unlawful killing should be left open to any future inquest as there could be further evidence that Mr Hogan was not suffering from insanity.

James Badenoch QC, appearing for Ms O'Connor and Mr Hogan, said it would be unsafe to rely on such evidence. It was unlikely to be "of sufficient quality" to undermine expert evidence from the two Greek psychiatrists who had found Mr Hogan was suffering from psychosis and not responsible for his actions.

He also warned that a finding of unlawful killing "without the very strongest of evidence" would not be in the interests of Mr Hogan's daughter Mia. He said such a verdict that her father "intended" to harm or kill her, as opposed to being mentally ill, could affect her growing into adulthood.

Mr Badenoch suggested a narrative verdict, with no reference to unlawful killing, could be substituted based on the evidence already available on how Liam met his death. He suggested that "no practical purpose or practical benefit" would come from "a further expensive, time consuming and, for all concerned, further harrowing reinvestigation of these matters."