Burdett returned late from club night at the Manukau Superstrike on Monday, March 23, 1992, and, after showering, was raped and battered to death.
Teina Pora spent 21 years in jail after he was wrongfully convicted of Burdett's rape and murder. His convictions were quashed by the Privy Council in 2015.
Tim McKinnel who worked for several years to show that Teina Pora had been wrongly convicted said the court application was encouraging.
"We have been asking for several years for the police to review the evidence and it is clear that they have done that and now filed documents in court seeking to retry Malcolm Rewa and that must be encouraging for Susan Burdett's family."
AUT University Professor of Law Warren Brookbanks told New Zealand Law Society in May lifting a stay of proceedings would be very unusual and seemed to be unprecedented in New Zealand.
"The effect of a stay is, of course, to forbid further steps being taken in those proceedings. It does not actually terminate them. Interestingly, there does not appear to be any statutory authority (at least that I am aware of) for lifting a stay."
Justice Minister Andrew Little announced earlier this month that Pora would receive an extra $1 million in compensation for the wrongful imprisonment.
The Privy Council quashed his convictions in 2015, and last year he received $2.52m and a government apology.
Little announced that Pora will receive an extra $988,099 as an inflation adjustment, bringing his total compensation package to $3,509,048.42.
