Matthew disappeared after being seen leaving the ARQ nightclub in Darlinghurst, Sydney during the early hours of September 23, 2007.
His former boyfriend Michael Atkins was the last known person to see him alive.
He was charged with murder in 2008 but acquitted in 2009.
Atkins had claimed that Matthew was alive and possibly living in Thailand.
However, he then led police to Matthew's skeletal remains in an unmarked grave in the Royal National Park in May 2017.
The remains were found beneath a cabbage-tree palm. The Levesons took the tree and planted it in their backyard.
Atkins told police he did not kill Matthew, but panicked and buried the body after he died of drug overdose.
DNA testing confirmed they belonged to Matthew.
Last December, the NSW Coroner has delivered an open finding into the 20-year-old's death.
Matthew's mother Faye Leveson said the family was forced to choose between justice and knowing the location of their son's body, after Atkins was granted immunity for the evidence he gave at the inquest, and immunity for perjury charges in return for leading police to Matthew's remains in the Royal National Park.
"Sadly, we were forced into this corner," Faye Leveson said. "And Matty comes out first every time."
