The 56-year-old is accused of killing Russell Hill and Carol Clay at their remote Wonnangatta campsite in Victoria's alpine region in March 2020.
But his legal team have made a bid to have key details in the prosecution case against him withheld from the public, including everything Lynn said to police after his arrest.
They argue it could pose a real and substantial risk of prejudice to the proper administration of justice.
Lynn has pleaded not guilty to murdering the couple.
Lead investigator Detective Senior Constable Brett Florence is expected to be cross-examined by Lynn's barrister Dermot Dann when a committal hearing continues in Melbourne Magistrates Court on Monday.
Among the evidence Det Florence and others are expected to be questioned about are Lynn's formal record of interview - taken over four days between November 22 and 25, 2021.
A statement Lynn made on July 14, 2020 and an hour-long recording of a conversation with police that same day are also sought by Lynn's legal team to be kept under wraps.
"The contents of these items of evidence are explosive, or may be seen as explosive in the eyes of readers or viewers," Mr Dann said during a hearing on Friday, referring to the potential for sensational reporting.
Lawyers for a number of media organisations challenged the suppression, which was not opposed by prosecutor John Dickie.
Magistrate Brett Sonnet questioned what the prejudice was in media waiting until a possible trial to report on the contents of Lynn's police interview.
But Corey Jankie, representing the ABC, said that wasn't the legal test for suppression orders, noting the legal assumption is that all material may be published.
Media reporting of the case demonstrated the substantial public interest in this case, he said.
Sam White, for the Nine Network, added that the law required only the "bare minimum" to be suppressed to prevent prejudice to an accused.
"We say the public has a right to know about the conduct of these proceedings," he said.
But Mr Dann argued to would be impossible to disentangle potentially prejudicial parts of the interview from more benign aspects, having regard to the length of the interview, and the approaches to the right to silence and to Lynn's legal advice.
He has previously described some of the evidence against Lynn as "highly inadmissible" and flagged future challenges to its inclusion in the case if Lynn is ordered to stand trial.
Mr Sonnet will hand down his decision on Monday.
Lynn is accused of murdering Mr Hill and Ms Clay at Wonnangatta on March 20, 2020 and of disposing of their bodies in remote bushland near Dargo.
Police also allege he set fire to the couple's Bucks Camp campsite to destroy potential evidence and that in May and/or November 2020 returned to their bodies and further tampered with their remains by "burning, dispersing and partially interring them".
Lynn was arrested on November 20, 2021. Remains later identified as those of the couple were discovered on November 29 that year.
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