Allan Schoenborn was found not criminally responsible because of mental illness in 2010 for the deaths of his daughter and two sons two years earlier.
In February, doctors at the hospital recommended the B.C. Review Board grant Schoenborn the escorted day outings, but the rest of the hearing was adjourned.
On Wednesday, when the hearing resumes, the Crown is expected to introduce a new witness who will counter the method that psychiatrists use to determine Schoenborn's risk to the public.
The aim is to lay a foundation to ask that Schoenborn be designated a high-risk offender under new federal legislation. Under the Not Criminally Responsible Reform Act, Schoenborn's case would have to be returned to the original trial judge to determine whether he is considered high risk.
While Schoenborn has had several review hearings, he has never been released from the psychiatric institute east of Vancouver.
Schoenborn's ex-wife, Darcie Clarke, and the mother of Kaitlynne, 10, Max, 8, and Cordon, 5, is strongly opposed to any freedom, according to Dave Teixeira, who speaks for Clarke and her family.
Teixeira says the family also wants Schoenborn to be labelled high risk.
"This is a man who does not deserve nor should be allowed out into society at this point, but rather should be focused on his treatment," said Teixeira.