Leading judges are being asked to block honeymoon murder suspect Shrien Dewani's extradition to South Africa until he is fit to stand trial.
Dewani is fighting removal to face trial over his wife Anni's death until he has recovered from mental health problems, including depression and post-traumatic stress disorder.
Members of Mrs Dewani's family are present for the hearing in London before the Lord Chief Justice Lord Thomas, Mr Justice Ouseley and Mr Justice Blake.
The hearing follows a decision by the High Court in October that there were outstanding legal issues the court must decide.
Dewani's lawyers have stressed at various hearings that he will be willing to defend himself at trial once he is fit to do so.
Today's proceedings at the High Court centre on two issues - the first relates to Dewani's status as ''an accused person'', and the second concerns whether it would be ''unjust and oppressive'' to extradite Dewani ''regardless of the prognosis'' of his mental condition.
Lord Thomas said that if Dewani's legal team won on the first issue, then Dewani "has to be discharged", but he "could be re-arrested subject to him remaining in this country when he becomes fit".
In discussion with Clare Montgomery QC, for Dewani, he said that if successful on the second issue, then "you have another adjournment, he remains where he is until better and then goes".
His lawyers say he is unfit to plead under English law and his "prognosis is not certain".
He is compulsorily detained in hospital under the Mental Health Act 1983 with his next annual review set for next May.
The first ground of appeal being addressed by the court asks whether a person who is unfit to plead is "an accused" for the purpose of the Extradition Act 2003 "if he is being extradited in circumstances where he may remain unfit to plead".
The second question being tackled by the judges is whether it is "unjust or oppressive to extradite a person who is agreed at the time of the determination to be unfit, whatever the prognosis".
In July, Chief Magistrate Howard Riddle ruled at Westminster Magistrates' Court that Dewani should be extradited and rejected his attempt to stay in the UK for further hospital treatment.
He said Dewani, from Bristol, was not fit to plead or stand trial at present, but there was evidence that he would receive the care he needed in South Africa.
Judge Riddle originally gave the go-ahead to Dewani's extradition in 2011 but had to reconsider the position after the High Court allowed his appeal in March last year.
Dewani is accused of ordering the killing of his new wife Anni, 28, who was shot as the couple travelled in a taxi on the outskirts of Cape Town in November 2010.
So far three men have been convicted over Mrs Dewani's death.
Last year South African Xolile Mngeni was convicted of premeditated murder for shooting her.
Prosecutors claimed he was a hitman hired by Dewani to kill his wife, which Dewani has consistently denied.
Taxi driver Zola Tongo was jailed for 18 years after he admitted his part in the killing and another accomplice, Mziwamadoda Qwabe, also pleaded guilty to murder and was handed a 25-year prison sentence.
Dewani's family have said that he remains committed to returning to South Africa ''when his health would permit a full trial and when appropriate protections are in place for his health and safety''.
The judges later reserved their decision which they said will be given in the New Year.