Kate and Gerry McCann are due to learn whether former Portuguese detective Goncalo Amaral has successfully overturned a ban on his book claiming their daughter Madeleine is dead.
Mr Amaral was the first head of the police investigation into the little girl's disappearance from Praia da Luz in Portugal in May 2007, shortly before her fourth birthday.
In July 2008 he published a book, Maddie: The Truth Of The Lie, which alleges that Madeleine died in her family's holiday flat and that her parents faked her abduction.
A judge granted the McCanns, both 41 and from Rothley, Leicestershire, a temporary injunction in September last year halting further sale or publication of Mr Amaral's book and a TV documentary he made about the case. The ex-policeman launched an appeal against the ban at the main civil court in the Portuguese capital Lisbon last month, calling a series of witnesses to support his claims.
After hearing closing arguments from lawyers last week, Judge Maria Gabriela Cunha Rodrigues reserved judgment.
Mr Amaral's lawyers argue the material in the book is contained in the official Portuguese police files for the investigation, many of which were made public when the case was shelved in August 2008.
The McCanns say their main motive for challenging the former policeman is the fear that people will stop looking for Madeleine if they think she is dead.
There will be a full trial at a later date on whether the injunction banning the book should be made permanent, the couple's Portuguese lawyer, Isabel Duarte, said.
The McCanns are also seeking 1.2 million euros (£1.08 million) in compensation for defamation in separate civil proceedings against Mr Amaral in Portugal.
The former detective has vowed to fight all the way to the European Court of Human Rights if he loses his case.
