The defence wants reporter Jana Winter to identify the law enforcement officials who told her that Holmes sent a notebook containing violent images to his psychiatrist before the 2012 attack.
Holmes' lawyers say whoever spoke to Winter violated a gag order and should be punished. They also say that officers might have lied when they denied under oath being Winter's sources, undermining their credibility as potential trial witnesses.
New York state's top court ruled in December that Winter did not have to testify in Colorado because she is protected by her home state's shield law, which says reporters do not have to identify confidential sources.
A Colorado court issued a subpoena for Winter's testimony, but because she is based in New York, that state's courts would have to enforce it.
Winter has said she would not identify the sources, even though the Colorado court could sentence her to jail for contempt of court for refusing.
Winter's attorney, Dori Hanswirth, didn't immediately return a call. Hanswirth previously expressed doubt that the Supreme Court would intervene.
Holmes has pleaded not guilty by reason of insanity to multiple counts of murder and attempted murder, and prosecutors are seeking the death penalty.
The identity of Winter's sources is not likely to influence the outcome of Holmes' case. But the defence's quest to identify them has dragged on for more than a year and drawn objections from Fox News and journalism organizations, who say it threatens reporters' ability to do their job.
Defence attorneys have acknowledged that Holmes was the shooter but say he was in the midst of a psychotic episode. The crucial question awaiting jurors is whether he was insane, which means unable to tell right from wrong under Colorado law. If the verdict is insanity, Holmes would be committed indefinitely to the state mental hospital and could not be executed.
Any evidence about his mental state — such as the notebook — could be important in persuading the jury. Authorities have confirmed Holmes sent the notebook, but the contents haven't been publicly confirmed.
Holmes' trial date is up in the air as attorneys work their way through scores of pretrial motions.
Holmes' lawyers revealed their plans to appeal to the Supreme Court at the end of a hearing on a defence motion to bar evidence from crime scene reconstruction experts. The defence said such evidence is unreliable.
Also pending is prosecutors' motion to have Holmes examined by psychiatric experts of their choosing. Holmes underwent a mandatory sanity evaluation at the state mental hospital last summer, but prosecutors claim the doctor who conducted the examination is biased.
The conclusions of that evaluation have not been made public.
Arapahoe County District Judge Carlos A. Samour Jr. has not said when he will rule on either the crime scene reconstruction evidence or the second evaluation.
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