Lawyers for the father of toddler Poppi Worthington, who was sexually abused shortly before she died, have claimed a coroner "overstepped the mark" at the inquest when he recorded the reasons for her death.
Former supermarket worker Paul Worthington, 50, believed to have been granted Legal Aid, took his case to the High Court, sitting in Manchester, where he was represented by high-profile barrister Leslie Thomas QC before three Law Lords.
A coroner has ruled that his daughter Poppi, aged 13 months, was anally penetrated in her father's double bed at the family home in Barrow-in-Furness in the early hours of December 12 2012.
The ruling mirrored the findings of two earlier High Court judgments.
During the inquest in January, Mr Worthington refused to answer questions about his daughter's death 252 times, so as not to incriminate himself.
He denies wrongdoing and has never been charged with any offence.
Cumbria Police botched their investigation into Poppi's death, losing vital evidence about her last hours.
However, at the conclusion of the inquest, senior coroner for Cumbria David Roberts ruled the child had been anally penetrated but that this did not cause her death.
He concluded that Poppi suffocated as she slept next to her father in an "unsafe sleeping environment".
Mr Worthington's lawyers applied for a judicial review of the coroner's record of inquest, the final two pages of his 87-page ruling, where the coroner must complete boxes in writing, detailing who and how the death occurred.
Mr Thomas said it was a "narrow" point but not "academic" and Mr Worthington was "entitled" to have the matter corrected.
He argued that the coroner was wrong to include the words "anally penetrated" in his record of inquest, also called an inquisition, which forms part of the official record, because, according to the law, this should only detail how the child died.
As the coroner had ruled that sexual abuse did not cause her death, those words should be taken out of the record of inquest.
Mr Thomas said:
We say this is an allegation of criminal conduct stuck in box three. Which should not be there. It doesn't touch on how this little girl died.
Samantha Leek QC, representing the coroner, said the anal penetration was part of the "wider factual matrix" giving rise to her death and Mr Thomas was trying to restrict the record of inquest to the immediate medical cause of death.
She came by her death because she is brought into bed, because she's brought into bed to be harmed.
Mr Justice Hickenbottom told the lawyers he would reserve judgment until a later date.
Mr Worthington is believed to be in hiding following publicity around his daughter's death.
A spokesman for the Legal Aid Agency said:
We cannot comment on individual cases.