BTW, Mary Mugford couldn´t have known whether Jeremy and June talked, I don´t believe she saw them together.
So many possibilities of exaggeration: Jeremy could have exaggerated this when talking to her, she could have exaggerated it in her testimony and finally, the author could have exaggerated for max effect. And it still doesn´t make Jeremy a murderer.
Seems like both children of the Bambers had problems with their mentally unstable mother, Sheila and Jeremy both.