The district attorney filed an 89-page brief asking justices to force attorney Richard Gammon to disclose any information about the UNC researcher's death in 2000.
Gammon represented Derril Willard, a man detectives believe played a role in Miller's poisoning.
Willard killed himself shortly after police searched his house. Investigators said Willard had a romantic relationship with Eric Miller's wife, Ann.
So far, no arrests have been made in the case.
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