The Massachusetts mother Lindsay Clancy will learn on Monday whether the murder charges against her can proceed, at a hearing where a judge is due to hear arguments on dismissing the case and on the timetable for any second trial.
The hearing in Plymouth comes after Clancy's first trial ended in deadlock, with jurors unable to reach a verdict.
Clancy, 36, is charged with first-degree murder over the deaths of Cora, Dawson and Callan in 2023.
Her defence is expected to argue that double jeopardy bars a second trial.
Her lawyers have said the mistrial declared on September 4 was unnecessary, and that a juror should have been questioned.
Prosecutors urged Sullivan in a filing last week to reject the dismissal bid.
The district attorney's office has not publicly detailed any plans for a retrial.
Sullivan told the parties at a September 29 hearing to be ready to discuss a retrial date.
The lone juror who blocked Lindsay Clancy from being acquitted was influenced by how “methodical” he thought she was while killing her three children, he said during an interview aired Thursday, his first since the trial ended with a deadlocked jury.
Michael Desronvil said he didn't have any doubt that Clancy was guilty, telling Fox News Channel's “Fox & Friends” that he was struck by her “thought process” during the killings.
Desronvil was the only juror who opposed clearing Clancy of criminal responsibility, siding with prosecutors who argued that she knew what she was doing.
Clancy's defense argued she was in the grip of a rare condition called postpartum psychosis when she strangled her kids.
“The focus has to be on the children,” Desronvil said. “They are the ones who don’t have anyone fighting for them.”
There was no dispute at the first trial that Clancy strangled the children with exercise bands before attempting suicide.

Her defence argued that postpartum psychosis meant she lacked criminal responsibility.
Jurors said one holdout blocked an insanity verdict that would have meant psychiatric hospital commitment.
Prosecutors could retry Clancy on first-degree murder or seek second-degree charges, which would allow a bench trial.
First-degree murder carries life without parole; second-degree allows parole.
Clancy's trial ended with a mistrial in early September and generated intense public interest, drawing attention to maternal mental health after childbirth.
She had mental health problems that included months of treatment and a stay in a psychiatric hospital before she killed the children and then tried to kill herself at her family's Massachusetts home in 2023.
Prosecutors have yet to decide whether they will put the case in front of a jury for a second time.
They could retry her on first-degree murder charges, seek lesser charges or drop the case entirely.
Clancy, a former labor and delivery nurse, remains charged with murder and is expected to stay at a psychiatric hospital until her case is resolved.
Her former husband Patrick Clancy, who discovered the children's bodies, has said in interviews that he forgives her.
If you are experiencing feelings of distress, or are struggling to cope, you can speak to the Samaritans, in confidence, on 116 123 (UK and ROI) or visit the Samaritans website to find details of your nearest branch.
If you are based in the USA, and you or someone you know needs mental health assistance right now, call or text 988, or visit 988lifeline.org to access online chat from the 988 Suicide and Crisis Lifeline. This is a free, confidential crisis hotline that is available to everyone 24 hours a day, seven days a week. If you are in another country, you can go to www.befrienders.org to find a helpline near you
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