Two psychologists testified Monday that John Battaglia is not mentally competent to be executed for killing his two young daughters in 2001 while their mother listened on the phone.
Battaglia, 60, was set to be put to death in March but won a last-minute reprieve from an appeals court.
His attorneys claim he is not mentally competent and cannot legally be put to death.
In a crime that drew national attention, Battaglia, an accountant, shot his daughters Faith, 9, and Liberty, 6, inside his Deep Ellum loft.
He then went to a tattoo parlor across the street and got tattoos of two flowers on his arm, one for each girl.
At Monday's hearing in a Dallas County courtroom, a third psychologist – hired by the court – testified Battaglia is competent under one legal standard but not another.
Battaglia's father, also named John, said outside the courtroom that he believes his son is mentally ill.
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"I have maintained from the beginning that he could not have been in his right senses to do it because he loved those girls without compromise," he said. "I mean, they were his whole life."
The elder Battaglia said his son is bipolar and suddenly changes his personality "from hot to cold."
"I miss my grandkids," he said. "But he's still my son. I can't separate that. I love him. I hate what he did, but I love him."
The hearing before Judge Robert Burns is set to resume at 9:30 a.m. Tuesday.