Follow CBSDFW.COM: Facebook | Twitter

DALLAS (CBSDFW.COM) – Testimony continues Tuesday in the competency hearing for John Battaglia. He is the North Texas father convicted of killing his young daughters 15 years ago while their mother, his ex-wife, listened helplessly on the phone.

READ MORE: Anna Police Investigating Crash That Killed 22-Year-Old Man

Battaglia’s life hinges on a court hearing happening Tuesday. The former Dallas accountant received a last-minute stay of execution in March when his attorney claimed him mentally incompetent. The State of Texas says it will not execute people with mental illnesses.

In court on Monday, three psychologists, who testified on behalf of both Battaglia and the state, said Battaglia has no rational understanding of what he did or why he should be put to death. They said the 61-year-old man’s delusions and personality disorders are real.

Battaglia’s father, who is also named John, attended the hearing and said the past 15 years in jail have not made his son any better. “He goes in and out in a conversation with me,” he said. “I’ve been at the jail visiting with him and suddenly he’d detect a tone in me he didn’t like and suddenly call for the jailer to take him back to his cell rather than talk to me.”

But a psychologist who testified Tuesday disagreed with his peers from the previous day. Based on interviews and conversations, Dr. James Womack believes that Battaglia is competent, and diagnosed the man with malingering disorder — claiming that Battaglia fabricated or exaggerated his symptoms of deluisions for personal gain.

Womack said that Battaglia is aware of what is happening around him. The psychologist described Battaglia as, “Calm, pleasant, smiled easily, acted rather congenially with me, showed no resistence to doing a social history. I requested of him that he help us stay focused — I do this with every defendent — stay focused on the topic of interest at the time. That prevents digressions. And he was very willing to do so.”

Photo of Faith and Liberty Battaglia, the daughters shot and killed by their father John Battaglia in 2001.

Photo of Faith and Liberty Battaglia, the daughters shot and killed by their father John Battaglia in 2001.

READ MORE: Texas Joins Oklahoma's Effort To Overturn McGirt

It was 2001 when Battaglia shot and killed Faith and Liberty, his 9- and 6-year-old daughters, in his Deep Ellum apartment. He had picked up the children from his ex-wife in the parking lot of a Park Cities shopping center for a court-ordered and scheduled visit. Battaglia was under a court order to stay away from his ex-wife’s Highland Park home.

Mary Jean Pearle listened helplessly as ex-husband murdered her two daughters. (credit: CBS 11 News)

Mary Jean Pearle listened helplessly as ex-husband murdered her two daughters. She was in court Tuesday for the competency hearing for John Battaglia. (credit: CBS 11 News)

Not long after the girls were given to Battaglia, he got his former wife, Mary Jean Pearle, on the phone. “Why do you want daddy to go to jail,” one of the girls reportedly said to Pearle over the phone. Seconds later, Pearle heard one of the children say “No, no, daddy,” then gunshots.

Hours later, Battaglia was found at a nearby tattoo shop getting two large red roses inked on his left arm to commemorate his daughters. When he walked outside, it took four officers to subdue and arrest him. A fully loaded revolver was found in his truck.

A jury sentenced him to die in 2002 but, in March of this year, the 5th U.S. Circuit Court of Appeals halted the execution about seven hours before he was to be taken to the Texas death chamber in Huntsville.

If the judge rules that Battaglia is competent, he will face a December 7 execution date. It’s something Battaglia’s father hopes doesn’t happen. “If I thought he was sane, I’d kill him myself. I can’t understand it.”

MORE NEWS: 2 Killed, 5 Injured After Drag Racer Slams Into Spectators An Texas Airport

(©2016 CBS Local Media, a division of CBS Radio Inc. All Rights Reserved. This material may not be published, broadcast, rewritten, or redistributed.)